Table of Contents

Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

39. Trinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 39.1 Recalling Gröbner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 39.2 Trinks’ Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 39.3 Gianni–Kalkbrener Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 39.4 An Ecumenic Notion of Solving . . . . . . . . . . . . . . . . . . . . . . . . . . 20 39.5 *Delassus–Gunther Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

40. Stetter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 40.1 Endomorphisms of an Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 40.2 Toward Auzinger–Stetter’s Theorem . . . . . . . . . . . . . . . . . . . . . . 33 40.3 Auzinger–Stetter: The Radical case . . . . . . . . . . . . . . . . . . . . . . . 36 40.4 *Möller: Endomorphisms and dual space . . . . . . . . . . . . . . . . . . 40 40.5 Möller–Stetter: the general case . . . . . . . . . . . . . . . . . . . . . . . . . . 46 40.6 The Univariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 40.7 Derogatoriness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 40.8 Stetter Algorithm via Grobnerian Technology . . . . . . . . . . . . . . 54 40.9 Stetter Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 40.10Derogatoriness and Allgemeine Coordinates . . . . . . . . . . . . . . . . 56

41. Macaulay IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 41.1 The resulatant of r forms in r variables . . . . . . . . . . . . . . . . . . . 65 41.2 Bézout’s Generating Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 41.3 Macaulay’s Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 41.4 The Extraneous Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 41.5 Macaulay’s Resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 41.6 Macaulay: The u-resultant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 41.7 Kronecker’s Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 41.8 Kronecker: the u-resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 41.9 Kronecker Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 41.10*Historical Intermezzo: from Bézout to Cayley . . . . . . . . . . . . . 96 41.11Dixon’s Resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 41.12Toward Cardinal’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 41.13Cardinal–Mourrain Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 41.14Mourrain: Proving Cardinal’s Conjecture . . . . . . . . . . . . . . . . . . 131 41.15Mourrain: A Gröbner-free Solver . . . . . . . . . . . . . . . . . . . . . . . . . 134

42. Lazard II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 42.1 Ritt: Characteristic sets for differential polynomial ideals …. 138 42.2 Ritt: Characteristic sets for polynomial ideals . . . . . . . . . . . . . . 147 42.3 Ritt’s and Wu’s Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 42.4 Lazard: Triangular sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 42.5 Admissible Lazard Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 42.6 Lazard’s Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 42.7 Ritt bases and Gröbner bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 42.8 Möller’s Zero-dimensional Solver . . . . . . . . . . . . . . . . . . . . . . . . . 175 42.9 Rouillier: Rational Univariate Representation . . . . . . . . . . . . . . 181

43. Lagrange II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 43.1 Representation of Groups as Permutation Groups. . . . . . . . . . . 193 43.2 Representation as Permutation Group of Roots . . . . . . . . . . . . . 200 43.3 Universal Lagrange Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 43.4 Cauchy modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 43.5 Resolvents and Polynomial Roots . . . . . . . . . . . . . . . . . . . . . . . . . 212 43.6 Lagrange resolvent and Galois group . . . . . . . . . . . . . . . . . . . . . . 216 43.7 Computing Galois groups of a polynomial . . . . . . . . . . . . . . . . . 221

44. Kronecker IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 44.1 Kronecker parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 44.2 Lifting Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 44.3 Newton–Hensel Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 44.4 Kronecker Package: Description . . . . . . . . . . . . . . . . . . . . . . . . . . 234 44.5 Kronecker Package: Lifting Step . . . . . . . . . . . . . . . . . . . . . . . . . . 236 44.6 Kronecker Package: Intersection Step . . . . . . . . . . . . . . . . . . . . . 237 44.7 Kronecker Package:Cleaning Step . . . . . . . . . . . . . . . . . . . . . . . . . 243 44.8 Genericity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 44.9 Complexity consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

45. Duval II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 45.1 Kronecker–Duval Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 45.2 Allgemeine Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 45.3 Kronecker Parametrization and Rational Universal Represen-

tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 45.4 Gröbner representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 45.5 Linear representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

4 Table of Contents

Setting

1. Let k be an infinite, perfect field, where, if p := char(k) ≠ 0, it is possible to extract pth roots and let k be the algebraic closure of k and Ω(k) the universal field over k.

Let us fix an integer value n and consider the polynomial ring

P := k[X_{1}, …, X_{n}]

and its k-basis

T := {X_{1}^{a_{1}} ⋯ X_{n}^{a_{n}} : (a_{1}, …, a_{n}) ∈ ℕ^{n}}.

For each d ∈ ℕ we will also set T_{d} := {t ∈ T : deg(t) = d}.

2. We also fix an integer value r ≤ n, set d := n − r and consider

the field K := k(V_{1}, …, V_{d}), its algebraic closure K and its universal field Ω(K) = Ω(k); the polynomial ring Q := K[Z_{1}, …, Z_{r}] and its K-basis W := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} : (a_{1}, …, a_{r}) ∈ ℕ^{r}}.

All the notation introduced will be applied also in this setting, just substituting everywhere n, k, P, T with, respectively r, K, Q, W.

3. Each polynomial f ∈ k[X_{1}, …, X_{n}] is a unique linear combination

f = ∑_{t∈T} c(f, t)t

of the terms t ∈ T with coefficients c(f, t) in k and can be uniquely decomposed, by setting

f_{δ} := ∑_{t∈T_{δ}} c(f, t)t, for each δ ∈ ℕ,

as f = ∑_{δ=0}^{d} f_{δ} where each f_{δ} is homogeneous, deg(f_{δ}) = δ, and f_{d} ≠ 0 so that d = deg(f).

4. Since, for each i, 1 ≤ i ≤ n,

P = k[X_{1}, …, X_{i−1}, X_{i+1}, …, X_{n}][X_{i}],

each polynomial f ∈ P can be uniquely expressed as

f = ∑_{j=0}^{D} h_{j}(X_{1}, …, X_{i−1}, X_{i+1}, …, X_{n})X_{i}^{j}, h_{D} ≠ 0,

and

deg_{X_{i}}(f) := deg_{i}(f) := D

denotes its degree in the variable X_{i}.

In particular (i = n)

f = ∑_{j=0}^{D} h_{j}(X_{1}, …, X_{n−1})X_{n}^{j}, h_{D} ≠ 0, D = deg_{n}(f);

the leading polynomial of f is Lp(f) := h_{D}, its trailing polynomial is Tp(f) := h_{0}.

5. Given a finite basis F := {f_{1}, …, f_{u}} ⊂ P, we denote

𝕀(F) := (F) := {∑_{i=1}^{u} h_{i}f_{i} : h_{i} ∈ P} ⊂ P

the ideal generated by F and

Z(F) := {a ∈ k^{n} : f(a) = 0, for all f ∈ F } ⊂ k^{n};

the algebraic variety consisting of each common root of all polynomials in F.

6. The support

supp(f) := {t ∈ T : c(f, t) ≠ 0}

of f being finite, once a term ordering^{1} < on T is fixed, f has a unique representation as an ordered linear combination of terms:

f = ∑_{i=1}^{s} c(f, t_{i})t_{i} : c(f, t_{i}) ∈ k \ 0, t_{i} ∈ T , t_{1} > ⋯ > t_{s}.

The maximal term of f is T(f) := t_{1}, its leading cofficient is lc(f) := c(f, t_{1}) and its maximal monomial is M(f) := c(f, t_{1})t_{1}.

7. For any set F ⊂ P we denote

• T_{<}{F } := {T(f) : f ∈ F };

1 A well-ordering < on T will be called a term ordering if it is a semigroup ordering.

• T_{<}(F) := {τT(f) : τ ∈ T , f ∈ F };

• N_{<}(F) := T \ T_{<}(F);

• k[N_{<}(F)] := Span_{k}(N_{<}(F))

and we will usually omit the dependence on < if there is no ambiguity.

8. Let < be a term ordering on T , I ⊂ P an ideal, and A := P/I. Since A ≅ k[N_{<}(I)], b for each f ∈ P, a unique

g := Can(f, I, <) = ∑_{t∈N_{<}(I)} γ(f, t, <)t,

the canonical form, such that

g ∈ k[N(I)] and f − g ∈ I.

9. For an ideal I ⊂ P,

I := ∩_{i=1}^{t}𝔮_{i}

denotes an irredundant primary representation in P; d := dim(I) its dimension and r := r(I) := n − d its rank; for each i, 𝔭_{i} := \sqrt{𝔮_{i}} is the associated prime.

10. For such ideal I we will re-enumerate and re-label the variables as

{X_{1}, …, X_{n}} = {V_{1}, …, V_{d}, Z_{1}, …, Z_{r}},

so that

I ∩ k[V_{1}, …, V_{d}] = (0), d := dim(I),

and we will wlog assume that the primaries are ordedered so that, for a suitable value 1 ≤ r ≤ t,

𝔮_{i} ∩ k[V_{1}, …, V_{d}] = (0), dim(𝔮_{i}) = d ⟺ i ≤ r

so that the ideal

J := Ik(V_{1}, …, V_{d})[Z_{1}, …, Z_{r}] = IQ

is zero dimensional and has, in Q, the irredundant primary representation

J := ∩_{i=1}^{r}𝔮_{i}Q

11. In general, when dealing witha 0-dimensional ideal, instead of

I ⊂ P = k[T ] = k[X_{1}, …, X_{n}]

we preferably use the notation

J ⊂ Q = K[W] = K[Z_{1}, …, Z_{r}].

12. For such 0-dimensional ideal J, with a slite abuse of notation, we still set A := Q/J and denote 𝔮_{i} its primary components in Q; we also assume

s := deg(J) = dim(A)

and we denote, for each f ∈ Q, [f] ∈ A its residue class modulo J and Φ_{f} the endomorphism

Φ_{f} : A → A, [g] ↦ [fg].

13. In terms of a K-basis q = {[q_{1}], …, [q_{s}]} of A so that A = Span_{K}(q), for each g ∈ Q, the Gröbner description of g is the unique (row) vector

Rep(g, q) := (γ(g, q_{1}, q), …, γ(g, q_{s}, q)) ∈ K^{s}

which satisfies

[g] = ∑_{j} γ(g, q_{j}, q)[q_{j}].

14. A Gröbner representation of J is the assignement of

• a K-linearly independent set q = {[q_{1}], …, [q_{s}]},

• the set M = M(q) := {(a_{lj}^{(h)}) ∈ K^{s^{2}}, 1 ≤ h ≤ r} of r square matrices

• s^{3} values γ_{ij}^{(l)} ∈ K

which satisfy

(1) Q/J ≅ Span_{K}(q),

(2) [Z_{h}q_{l}] = ∑_{j} a_{lj}^{(h)}[q_{j}], for each l, j, h, 1 ≤ l, j ≤ s, 1 ≤ h ≤ r,

(3) [q_{i}q_{j}] = ∑_{l} γ_{ij}^{(l)}[q_{l}] for each l, j, h, 1 ≤ i, j, l ≤ s.

A Gröbner representation is called a linear representation iff q = N_{<}(J) wrt a term ordering <.

15. For the 0-dimensional ideal J ⊂ Q with the irredundant primary representation J = ⋂_{i=1}^{r} 𝔮_{i} in Q, we set, for each i, 1 ≤ i ≤ r,

• 𝔪_{i} = \sqrt{q_{i}}, the associated maximal prime,

• K_{i} := Q/𝔪_{i}, K ⊂ K_{i} ⊂ K,

• Q_{i} := K_{i}[Z_{1}, …, Z_{r}],

• the irredundant primary representations 𝔮_{i} = ∩_{j=1}^{r_{i}}q_{ij} and 𝔪_{i} = ∩_{j=1}^{r_{i}}m_{ij} in Q_{i},

• the roots b_{ij} := (b_{1}^{(ij)}, …, b_{r}^{(ij)}) ∈ K_{i}^{r} ⊂ K^{r}, 1 ≤ j ≤ r_{i},

• d_{ij} := mult(b_{ij}, J) = deg(q_{ij}) for each j, 1 ≤ j ≤ r_{i},

which satisfy:

(1) m_{ij} = (Z_{1} − b_{1}^{(ij)}, …, Z_{r} − b_{r}^{(ij)}),

(2) the b_{ij}s, 1 ≤ j ≤ r_{i}, are K-conjugate for each i,

(3) up to a renumeration, \sqrt{q_{ij}} = m_{ij}, (4) 𝔪_{i} = m_{ij} ∩ Q,

(5) 𝔮_{i} = q_{ij} ∩ Q,

(6) for each j, l, 1 ≤ j, l ≤ r_{i}, d_{ij} = d_{il} =: d_{i},

(7) r_{i} = deg(𝔪_{i}) = [K_{i} : K],

(8) deg(𝔮_{i}) = d_{i}r_{i},

(9) J = ∩_{i=1}^{r} ∩_{j=1}^{r_{i}} q_{ij}, \sqrt{J} = ∩_{i=1}^{r} ∩_{j=1}^{r_{i}} m_{ij} are the irredundant primary representations in K[Z_{1}, …, Z_{r}],

(10) Z(J) = {b_{ij} : 1 ≤ i ≤ r, 1 ≤ j ≤ r_{j}},

(11) ∑_{i=1}^{r} d_{i}r_{i} = s.

16. With the notation above the ideal J has s := ∑_{i=1}^{r} r_{i} roots which we will also denote as

Z(J) = {α_{1}, …, α_{s}} ⊂ K^{r}, α_{i} = (a_{1}^{(i)}, …, a_{r}^{(i)}).

For each such root α_{i} we set

• 𝔪_{α_{i}} = (Z_{1} − a_{1}^{(i)}, …, Z_{r} − a_{r}^{(i)}),

• 𝔮_{i} the 𝔪_{α_{i}}-primary component of J, so that J = ∩_{i=1}^{s}𝔮_{i} in K ⊗_{K} Q;

• s_{i} := mult(α_{i}, J) = deg(𝔮_{i}) the multiplicity in J of α_{i} so that s = ∑_{i=1}^{s} s_{i}.

17. A linear form Y := ∑_{h=1}^{r} c_{h}Z_{h} is said an allgemeine coordinate for the 0-dimensional ideal J iff

(a) there are polynomials g_{i} ∈ K[Y], 0 ≤ i ≤ n, g_{0} monic, deg(g_{i}) < deg(g_{0}), such that

G := (g_{0}(Y), Z_{1} − g_{1}(Y), Z_{2} − g_{2}(Y), …, Z_{r} − g_{r}(Y))

is the reduced Gröbner basis of the ideal

J^{+} := J + (Y − ∑_{h} c_{h}Z_{h}) ⊂ K[Y, Z_{1}, …, Z_{r}]

w.r.t. the lex ordering induced by Y < Z_{1} < … < Z_{r};

with the present notation, this condition implies, among the others, that (Corollary 34.4.6)

(b) Q/J ≅ K[Y]/g_{0}(Y)

(c) for each i, 1 ≤ i ≤ s, β_{i} := ∑_{h=1}^{r} c_{h}a_{h}^{(i)} is a root of g_{0} with multiplicity s_{i} and

(d) a_{j}^{(i)} = g_{j}(β_{i}) for each i, 1 ≤ i ≤ s, and each j, 1 ≤ j ≤ r,

(e) g_{0}(Y) = ∏_{i=1}^{r}(Y − β_{i})^{s_{i}};

(f) f ∈ J ⟺ Rem(f(g_{1}(Y), …, g_{r}(Y)), g_{0}(Y)) = 0.

Moreover, there is a Zarisky open set U ⊂ K^{n} such that Y := ∑_{h=1}^{r} c_{h}Z_{h} is an allgemeine coordinate for J iff (c_{1}, …, c_{r}) ∈ U.

18. Given the polynomial ring P := k[X_{1}, …, X_{n}] and its monomial k-basis T we introduce n futher variables Y_{1}, …, Y_{n} and we denote

• P_{Y} := k[Y_{1}, …, Y_{n}] and T_{Y} its corresponding monomial k-basis;

• P_{⊗} := P ⊗ Q = k[X_{1}, …, X_{n}, Y_{1}, …, Y_{n}], and T_{⊗} its corresponding monomial k-basis T_{⊗} := {τ ⊗ ω : τ ∈ T , ω ∈ T_{Y} };

• for each i, 0 ≤ i ≤ n, we use the notation h(X_{i}) to denote the polynomial

h(X_{i}) := h(Y_{1}, …, Y_{i}, X_{i+1}, …, X_{n}) for each h(X_{1}, …, X_{n}) ∈ P;

in particular h(X_{0}) = h(X_{1}, …, X_{n}) and h(X_{n}) = h(Y_{1}, …, Y_{n}).

• for an ideal I = 𝕀(f_{1}, …, f_{s}) ⊂ P with a slight abuse of notation we denote I also the ideal in P_{Y} generated by {f_{1}(Y_{1}, …, Y_{n}), …, f_{n}(Y_{1}, …, Y_{n})} and A := P_{Y} /I; thus we have also

A ⊗_{k} A = P_{⊗}/𝕀 (f_{i}(X_{1}, …, X_{n}), f_{i}(Y_{1}, …, Y_{n}), 1 ≤ i ≤ n) ;

• finally we denote I_{X} := I ⊗ P_{Y} ⊂ P_{⊗} and I_{Y} := P ⊗ I ⊂ P_{⊗}.

39. Trinks

The first paper applying Buchberger’s Algorithm being Trinks proposal of an algorithm for solving polynomial equations systems, Trinks’ Algorithm is the natural choice for opening this section on algebraic solving.

Trinks’ Algorithm essentially is an effective reformulation of Gröbner’s proof of Hilbert’s Nullstellensatz: given a 0-dimensional ideal J ⊂ Q := K[Z_{1}, …, Z_{r}], iteratively Trinks’ Algorithm, for each roots α ∈ K^{i−1} of J ∩ K[Z_{1}, …, Z_{i−1}], computes and solves gcd(h(α, Z_{i}) : h ∈ G_{i}) ∈ K[Z_{i}] where G_{i} denotes a basis of the ideal J ∩ K[Z_{1}, …, Z_{i}]; the rôle of Gröbner bases consits in allowing to compute such basis of the elimination ideals.

The main improvement to Trinks’ Algorithm, a part from the use of FGLM in order to efficiently deduce the needed lex Gröbner basis of J, is Gianni–Kalkbrener’s proposal of using their Theorem; the evaluation at α of all polynomials in G_{i} and the computation of their gcd is thus reduced to the evaluation at α of the leading polynomials of some elements in G_{i} and of the first element whose leading polynomial is not vanishing at α.

After recalling the basic tools provided by Gröbner bases w.r.t. solving (Section 39.1) I present Trinks’ (Section 39.2) and Gianni–Kalkbrener’s Algorithm (Section 39.3) concluding with some comments which aim to read these algorithms in the setting of Kronecker–Duval Philosophy (Section 39.4). Finally (Section 39.5) I discuss a solver dated 1913 which already explicitly applies the main property of the lex term ordering and anticipates Macaulay’s Lemma.

39.1 Recalling Gröbner

Let us consider

an infinite, perfect^{1} field k, where, if p := char(k) ≠ 0, it is possible to

extract pth roots, the algebraic closure k of k, the universal field Ω(k) over k (Definition 9.4.1);

1 While the techniques discussed here apply in this general setting we are mainly thinking of the case k = ℚ, k := ℂ; on the other side technically we need to (and we can) solve over ℚ(V_{1}, …, V_{d}).

the polynomial ring P := k[X_{1}, …, X_{n}], its k-basis T := {X_{1}^{a_{1}} ⋯ X_{n}^{a_{n}} : (a_{1}, …, a_{n}) ∈ ℕ^{n}}; an ideal^{2} I := (F) := 𝕀(F) := {∑_{i=1}^{u} h_{i}f_{i} : h_{i} ∈ P} ⊂ P given by a finite basis F := {f_{1}, …, f_{u}} ⊂ P, the algebraic affine variety

Z(I) := {a ∈ k^{n} : f(a) = 0, for each f ∈ F } ⊂ k^{n}.

Each polynomial f ∈ k[X_{1}, …, X_{n}] is therefore a unique linear combination

f = ∑_{t∈T} c(f, t)t

of the terms t ∈ T with coefficients c(f, t) in k; the support

supp(f) := {t ∈ T : c(f, t) ≠ 0}

of f being finite, once a term ordering^{3} < on T is fixed, f has a unique representation as an ordered linear combination of terms:

f = ∑_{i=1}^{s} c(f, t_{i})t_{i} : c(f, t_{i}) ∈ k \ 0, t_{i} ∈ T , t_{1} > ⋯ > t_{s};

the maximal term of f is T(f) := t_{1}, its leading coefficient is lc(f) := c(f, t_{1}) and its maximal monomial is M(f) := c(f, t_{1})t_{1}.

For any set F ⊂ P we denote

• T_{<}{F } := {T(f) : f ∈ F };

• T_{<}(F) := {τT(f) : τ ∈ T , f ∈ F };

• N_{<}(F) := T \ T_{<}(F);

• k[N_{<}(F)] := Span_{k}(N_{<}(F))

and we will usually omit the dependence on < if there is no ambiguity. Recall that

Definition 39.1.1 (Buchberger). A subset G ⊂ I will be called a Gröbner basis of I w.r.t. < if T(G) = T{I}, id est T{G} generates the monomial ideal T(I) = T{I}.

For each f ∈ P the canonical form of f w.r.t. I is the unique polynomial

g := Can(f, I, <) = ∑_{t∈N(I)} γ(f, t, <)t ∈ k[N(I)]

such that f − g ∈ I.

2 All over the book I will use the notation 𝕀(F) ⊂ R in order to denote the ideal generated by the basis F in the ring R; when there is no ambiguity R will be not specified.

3 Recall that (cf. Definition 22.1.2) a well-ordering < on T will be called a term ordering if it is a semigroup ordering, id est

t_{1} < t_{2} ⟹ tt_{1} < tt_{2}, for each t, t_{1}, t_{2} ∈ T .

Let us fix any term-ordering < on T and let us compute a Gröbner basis G ⊂ I of I w.r.t. <.

Then it holds (cf. Remark 27.12.4)

• Z(I) = ∅ ⟺ 1 ∈ I ⟺ 1 ∈ G;

• Z(I) is infinite iff N(I) is an infinite dimensional k-vector space iff there exists i such that for each d ∈ ℕ : X_{i}^{d} ∉ T(G) = T (I);

• Z(I) is finite iff N(I) is finite iff for each i there exists d_{i} ∈ ℕ : X_{i}^{d_{i}} ∈ T(G) ⊂ T(I); moreover, in this case and under the assumption that I is radical, we have #Z(I) = #N(I).

Kredel–Weispfenning algorithm (cf. Corollary 27.11.9) allows to deduce from T(I) the dimension d := dim(I), the rank r := n − d := r(I) of I and a maximal set of independent variables (cf. Definition 27.11.4) {X_{i_{1}}, …, X_{i_{d}}} so that I ∩ k[X_{i_{1}}, …, X_{i_{d}}] = (0).

Then, we can re-enumerate and re-label the variables as

{X_{1}, …, X_{n}} = {V_{1}, …, V_{d}, Z_{1}, …, Z_{r}}, {X_{i_{1}}, …, X_{i_{d}}} = {V_{1}, …, V_{d}},

so that

I ∩ k[V_{1}, …, V_{d}] = (0)

and consider

the field K := k(V_{1}, …, V_{d}), its algebraic closure K and its universal field Ω(K) = Ω(k); the polynomial ring Q := K[Z_{1}, …, Z_{r}], its K-basis W := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} : (a_{1}, …, a_{r}) ∈ ℕ^{r}}; the zero-dimensional ideal J := I^{e} := IK[Z_{1}, …, Z_{r}] and the unmixed ideal J^{c} := J ∩ P.

Then, if I = ∩_{i=1}^{t}𝔮_{i} denotes any irredundant primary representation in P, and we wlog assume that the primaries are ordedered so that, for a suitable value 1 ≤ r ≤ t,

{X_{i_{1}}, …, X_{i_{d}}} is a maximal set of independent variables for 𝔮_{i} ⟺ i ≤ r,

then Corollary 27.5.19 grants that

J := I^{e} = ⋂_{i=1}^{r} 𝔮_{i}^{e} = ⋂_{i=1}^{r} 𝔮_{i}Q

is an irredundant primary representation in Q and

J^{c} := I^{ec} = ⋂_{i=1}^{r} 𝔮_{i} ⊂ P

is an irredundant primary representation.

Moreover, the (GTZ, ARGH, CCC)-schemes (Chapter 35) allow to compute unmixed ideals 𝔞_{j} ⊂ P giving a decomposition

\sqrt{I} = \sqrt{J^{c}} ⋂ (⋂_{j} \sqrt{𝔞_{j}}) .

Thus solving the ideal I ⊂ P is reduced, via Gröbner technique, to solving each unmixed (GTZ, ARGH, CCC)-component and solving each such component is reduced to solving the related zero-dimensional extension ideal.

39.2 Trinks’ Algorithm

Thus we are reduced to consider a zero-dimensional ideal

J ⊂ Q := K[Z_{1}, …, Z_{r}]

which we assume to be given via a Gröbner basis G_{≺} w.r.t. the lexicographical ordering ≺ induced on W by Z_{1} ≺ Z_{2} ≺ ⋯ ≺ Z_{r}:

Z_{1}^{a_{1}} … Z_{r}^{a_{r}} ≺ Z_{1}^{b_{1}} … Z_{r}^{b_{r}} ⟺ exists j : a_{j} < b_{j} and a_{i} = b_{i} for i > j.

Then, if we denote, for i, 1 ≤ i < r,

J_{i} := J ∩ K[Z_{1}, …, Z_{i}], π_{i} : K^{r} → K^{i} the canonical projection π_{i}(a_{1}, …, a_{r}) = (a_{1}, …, a_{i}), G_{i} := G_{≺} ∩ K[Z_{1}, …, Z_{i}],

we have, for each i

(1) Z(J_{i}) = π_{i}(Z(J)) = {(a_{1}, …, a_{i}) : (a_{1}, …, a_{r}) ∈ Z(J)},

(2) G_{i} is the reduced lexicographical Gröbner basis of J_{i} (Corollary 26.2.4).

In particular, there is a unique polynomial f(Z_{1}) ∈ K[Z_{1}], such that

J_{1} = (f) and {f} = G_{≺} ∩ K[Z_{1}].

For each α := (a_{1}, …, a_{i−1}) ∈ K^{i−1}, denote Φ_{α} : K[Z_{1}, …, Z_{i}] → K[T] the projection defined by

Φ_{α}(f) = f(a_{1}, …, a_{i−1}, T) for each f ∈ K[Z_{1}, …, Z_{i}].

Theorem 39.2.1 (Trinks). Let α := (a_{1}, …, a_{i−1}) ∈ Z(J_{i−1}) and let f ∈ K[T] be a generator of the principal ideal Φ_{α}(J_{i}) ⊂ K[T]. Then, for each b ∈ K

(a_{1}, …, a_{i−1}, b) ∈ Z(J_{i}) ⟺ f(b) = 0.

Proof. Let h(Z_{1}, …, Z_{i}) ∈ J_{i} be any polynomial such that

f(T) = Φ_{α}(h) = h(a_{1}, …, a_{i−1}, T).

Then

(a_{1}, …, a_{i−1}, b) ∈ Z(J_{i}) ⟹ f(b) = h(a_{1}, …, a_{i−1}, b) = 0.

Conversely for any g(Z_{1}, …, Z_{i}) ∈ J_{i}, Φ_{α}(g) ∈ Φ_{α}(J_{i}), so that

g(a_{1}, …, a_{i−1}, b) = Φ_{α}(g)(b) = 0 for each g ∈ J_{i}
and (a_{1}, …, a_{i−1}, b) ∈ Z(J_{i}).  �

Algorithm 39.2.2 (Trinks). Trinks’ Algorithm (Figure 39.1) for ’solving’ a zero-dimensional ideal is based on the Theorem above and consists in iteratively computing Z(J_{i}) by ’solving’, for each α ∈ Z(J_{i−1}), the univariate polynomial generating the principal ideal Φ_{α}(J_{i}).

Fig. 39.1. Trinks’ Algorithm

Z := Solve(F, L)

where F := (f_{1}, …, f_{u}) ⊂ Q := K[Z_{1}, …, Z_{r}], L ⊃ K is a field extension of K, J ⊂ Q is the zero-dimensional ideal generated by F, Z := {α_{1}, …, α_{s}} = Z(J) ∩ L^{r}.

Compute the reduced lexicographical Gröbner basis G of (f_{1}, …, f_{u}).

Let p(Z_{1}) be the unique element in G ∩ K[Z_{1}],

Z_{1} := {a ∈ L : p(a) = 0}.

For i = 2..r do

Z_{i} := ∅;
For each (a_{1}, …, a_{i−1}) ∈ Z_{i−1} do
H := {g(a_{1}, …, a_{i−1}, Z_{i}) : g ∈ G_{i} \ G_{i−1}},
p := gcd(H),
Z := {a ∈ L : p(a) = 0},
Z_{i} := Z_{i} ∪ {(a_{1}, …, a_{i−1}, a) : a ∈ Z}.

Z := Z_{r}

Example 39.2.3. To illustrate Trinks’ Algorithm let us consider the zerodimensional ideal J ⊂ ℚ[Z_{1}, Z_{2}, Z_{3}] discussed in Example 33.2.6 whose lex Gröbner basis^{4} is G := {g_{i}, 1 ≤ i ≤ 7}, where (Examples 33.5.1 and 33.5.2)

4 The leading polynomial (page 17) Lp(g_{i}) is indicated in bold.

g_{1} := 1Z_{1}^{3} − 3Z_{1}^{2} + 2Z_{1},
g_{2} := (Z_{1}^{2} − Z_{1})Z_{2},
g_{3} := Z_{1}Z_{2}^{2} − Z_{1}Z_{2},
g_{4} := 1Z_{2}^{3} − 3Z_{2}^{2} + 2Z_{2},
g_{5} := (Z_{1}^{2} − 3Z_{1} + 2)Z_{3} − 3Z_{2}^{2} − 6Z_{2}Z_{1} + 9Z_{2} − Z_{1}^{2} + 3Z_{1} − 2,
g_{6} := (Z_{2} + Z_{1} − 2)Z_{3} + 3Z_{2}^{2} + Z_{2}Z_{1} − 7Z_{2} − 2Z_{1}^{2} + 3Z_{1} + 2,
g_{7} := (Z_{1} − 2)Z_{3}^{2} − 4Z_{3}Z_{1} + 8Z_{3} − 15Z_{2}^{2} − 30Z_{2}Z_{1} + 45Z_{2} + 3Z_{1} − 6,
g_{8} := 1Z_{3}^{3} − 3Z_{3}^{2} + 3Z_{3}Z_{1} − 4Z_{3} − 3Z_{2}^{2} − 6Z_{2}Z_{1} + 9Z_{2} − 3Z_{1} + 6,

and whose roots are Z(J) = {b_{j}, 1 ≤ j ≤ 9} where

b_{1} = (0, 0, 1) b_{2} = (0, 1, −2) b_{3} = (2, 0, 2)
b_{4} = (0, 2, −2) b_{5} = (1, 0, 3) b_{6} = (1, 1, 3)
b_{7} = (1, 1, 1) b_{8} = (2, 0, 1) b_{9} = (2, 0, 0).

Then we have:

p(Z_{1}) := g_{1}, Z_{1} := {0, 1, 2};

α = (0) : Φ_{α}(g_{2}) = Φ_{α}(g_{3}) = 0; Φ_{α}(g_{4}) = T^{3} − 3T^{2} + 2T;

Z := {0, 1, 2}, Z_{2} := {(0, 0), (0, 1), (0, 2)};

α = (1) : Φ_{α}(g_{2}) = 0; Φ_{α}(g_{3}) = T^{2} − T; Φ_{α}(g_{4}) = T^{3} − 3T^{2} + 2T;

gcd(Φ_{α}(g_{3}), Φ_{α}(g_{4})) = T^{2} − T; Z := {0, 1}, Z_{2} := {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)};

α = (2) : Φ_{α}(g_{2}) = 2T; Φ_{α}(g_{3}) = 2T^{2} − 2T; Φ_{α}(g_{4}) = T^{3} − 3T^{2} + 2T;

gcd(Φ_{α}(g_{i}), 2 ≤ i ≤ 4) = T; Z := {0}, Z_{2} := {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)};

α = (0, 0) : Φ_{α}(g_{5}) = 2T − 2; Φ_{α}(g_{6}) = −2T + 2;

Φ_{α}(g_{7}) = −2T^{2} + 8T − 6; Φ_{α}(g_{8}) = T^{3} − 3T^{2} − 4T + 6; gcd(Φ_{α}(g_{i}), 5 ≤ i ≤ 8) = T − 1; Z := {1}, Z_{2} := {(0, 0, 1)};

α = (0, 1) : Φ_{α}(g_{5}) = 2T + 4; Φ_{α}(g_{6}) = −T − 2;

Φ_{α}(g_{7}) = −2T^{2} + 8T + 24; Φ_{α}(g_{8}) = T^{3} − 3T^{2} − 4T + 12; gcd(Φ_{α}(g_{i}), 5 ≤ i ≤ 8) = T + 2; Z := {−2}, Z_{2} := Z_{2} ∪ {(0, 1, −2)};

α = (0, 2) : Φ_{α}(g_{5}) = 2T + 4; Φ_{α}(g_{6}) = −T − 2;

Φ_{α}(g_{7}) = −2T^{2} + 8T + 24; Φ_{α}(g_{8}) = T^{3} − 3T^{2} − 4T + 12; gcd(Φ_{α}(g_{i}), 5 ≤ i ≤ 8) = T + 2; Z := {−2}, Z_{2} := Z_{2} ∪ {(0, 2, −2)};

α = (1, 0) : Φ_{α}(g_{5}) = 0; Φ_{α}(g_{6}) = −T + 3;
Φ_{α}(g_{7}) = −T^{2} + 4T − 3; Φ_{α}(g_{8}) = T^{3} − 3T^{2} − T + 3;
gcd(Φ_{α}(g_{i}), 6 ≤ i ≤ 8) = T − 3;
Z := {3}, Z_{2} := Z_{2} ∪ {(1, 0, 3)};
α = (1, 1) : Φ_{α}(g_{5}) = 0; Φ_{α}(g_{6}) = 0;
Φ_{α}(g_{7}) = −T^{2} + 4T − 3; Φ_{α}(g_{8}) = T^{3} − 3T^{2} − T + 3;
gcd(Φ_{α}(g_{i}), 7 ≤ i ≤ 8) = T^{2} − 4T + 3;
Z := {1, 3}, Z_{2} := Z_{2} ∪ {(1, 1, 3), (1, 1, 1)};
α = (2, 0) : Φ_{α}(g_{5}) = Φ_{α}(g_{6}) = Φ_{α}(g_{7}) = 0; Φ_{α}(g_{8}) = T^{3} − 3T^{2} − T + 3;
gcd(Φ_{α}(g_{i}), 7 ≤ i ≤ 8) = T^{3} − 3T^{2} − T + 3;
Z := {0, 1, 2}, Z_{2} := Z_{2} ∪ {(2, 0, 0), (2, 0, 1), (2, 0, 2)}.

39.3 Gianni–Kalkbrener Algorithm

A part from the FGLM-proposal of indirectly producing the needed lexicographical Gröbner basis via linear algebra from the Gröbner bases wrt an easier-to-compute termordering, the most relevant improvement on Trinks’ Algorithm is based on the deeper analysis performed by Gianni and Kalkbrener on the structure of the lexicographical Gröbner basis of a zerodimensional ideal.

Remarking that each polynomial f ∈ K[Z_{1}, …, Z_{i}] can be uniquely expressed as

f = ∑_{j=0}^{D} h_{j}(Z_{1}, …, Z_{i−1})Z_{i}^{j}, h_{D} ≠ 0,

we recall that the degree of f in the variable Z_{i} is denoted

deg_{Z_{i}}(f) := deg_{i}(f) := D

and that Lp(f) := h_{d} is named the leading polynomial of f and we observe that, for the lexicographical ordering ≺, we have T(f) = T(Lp(f))Z_{i}^{deg_{i}(f)}.

We also denote, for each i, 1 ≤ i ≤ r, δ ∈ ℕ,

G_{iδ} := {g ∈ G, g ∈ K[Z_{1}, …, Z_{i}], deg_{i}(g) ≤ δ}

and remark that each G_{iδ} is a section of both G_{iδ+1} and G_{i} and that hold the obvious inclusions

G_{11} ⊆ G_{12} ⊆ … ⊆ G_{1} ⊆ … ⊆ G_{i−1} ⊆ … ⊆ G_{iδ} ⊆ G_{iδ+1} ⊆ … ⊆ G_{i} ⊆ …

For each i, 1 ≤ i ≤ r, δ ∈ ℕ, and each F ⊂ Q, we also denote

Lp_{iδ}(F) := {Lp(g), g ∈ F ∩ K[Z_{1}, …, Z_{i}], deg_{i}(g) ≤ δ}.

Theorem 39.3.1 (Gianni—Kalkbrener). Let J ⊂ Q be an ideal, ≺ be the lexicographical ordering induced by Z_{1} ≺ ⋯ ≺ Z_{r}.

Let G := {g_{1}, …, g_{v}} be a Gröbner basis of J w.r.t. ≺, enumerated in such a way that

T(g_{1}) ≺ T(g_{2}) ≺ … ≺ T(g_{v−1}) ≺ T(g_{v}).

Then with the notation above:

(1) for each i, i ≤ r, G_{i} is a Gröbner basis of J_{i};

(2) for each i, 1 ≤ i ≤ r, δ ∈ ℕ, Lp_{iδ}(G) is a Gröbner basis of Lp_{iδ}(J);

(3) for each i, 1 ≤ i ≤ r and each α := (b_{1}, …, b_{i−1}) ∈ Z(J_{i−1}), denoting σ the minimal value such that Φ_{α}(Lp(g_{σ})) ≠ 0 and j, δ the value such that

g_{σ} = Lp(g_{σ})Z_{j}^{δ+1} + ⋯ ∈ K[Z_{1}, …, Z_{j}] \ K[Z_{1}, …, Z_{j−1}]

it holds

(a) j = i,

(b) for each g ∈ G_{i−1}, Φ_{α}(g) = 0,

(c) for each g ∈ G_{iδ}, Φ_{α}(g) = 0,

(d) Φ_{α}(g_{σ}) = gcd (Φ_{α}(g) : g ∈ G_{i}) ∈ K[T],

(e) for each b ∈ K,

(b_{1}, …, b_{i−1}, b) ∈ Z(J_{i}) ⟺ Φ_{α}(g_{σ})(b) = 0.
Proof. cf. Section 26.2 and 34.6.  �

Algorithm 39.3.2 (Gianni–Kalkbrener). Gianni–Kalkbrener improvement to Trinks’ Algorithm allows to avoid, for each α := (a_{1}, …, a_{i−1}) ∈ Z_{i−1}, both the complete evaluation Φ_{α}(g) of all g ∈ G_{i} \ G_{i−1} and the computation of their gcd, reducing this step to the evaluation of the leading polynomials of a suitable subset of such elements (Figure 39.2).

Example 39.3.3. In Example 39.2.3, Gianni–Kalkbrener Algorithm computes

Z_{1} := {0, 1, 2};
α = (0) : Φ_{α}(Lp(g_{2})) = Φ_{α}(Lp(g_{3})) = 0, Φ_{α}(Lp(g_{4})) = 1;
Φ_{α}(g_{4}) = T^{3} − 3T^{2} + 2T;
Z := {0, 1, 2}, Z_{2} := {(0, 0), (0, 1), (0, 2)};
α = (1) : Φ_{α}(Lp(g_{2})) = 0; Φ_{α}(Lp(g_{3})) = 1;
Φ_{α}(g_{3}) = T^{2} − T;
Z := {0, 1}, Z_{2} := {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)};
α = (2) : Φ_{α}(Lp(g_{2})) = 2, Φ_{α}(g_{2}) = 2T;
Z := {0}, Z_{2} := {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)};

Fig. 39.2. Trink’s Algorithm; Gianni—Kalkbrener improvement

Z := Solve(F, L)

where F := (f_{1}, …, f_{u}) ⊂ Q := K[Z_{1}, …, Z_{r}], L ⊃ K is a field extension of K, J ⊂ Q is the zero-dimensional ideal generated by F, Z := {α_{1}, …, α_{s}} = Z(J) ∩ L^{r}.

Compute the reduced lexicographical Gröbner basis G of (f_{1}, …, f_{u}).

Sort G := {g_{1}, …, g_{v}} by increasing maximal terms.

Z_{1} := {a ∈ L : g_{1}(a) = 0},

%% g_{1} is the unique element in G ∩ K[Z_{1}].

For i = 2..r do

Z_{i} := ∅;
g := min(g ∈ G_{i} \ G_{i−1}).
For each (a_{1}, …, a_{i−1}) ∈ Z_{i−1} do
h := g,
While Lp(h)(a_{1}, …, a_{i−1}) = 0 do h := Next(h, G),
p := h(a_{1}, …, a_{i−1}, Z_{i}),
%% p = gcd(H) for H := {g(a_{1}, …, a_{i−1}, Z_{i}) : g ∈ G_{i} \ G_{i−1}},
Z := {a ∈ L : p(a) = 0},
Z_{i} := Z_{i} ∪ {(a_{1}, …, a_{i−1}, a) : a ∈ Z}.

Z := Z_{r}

α = (0, 0) : Φ_{α}(Lp(g_{5})) = 2, Φ_{α}(g_{5}) = 2T − 2;
Z_{2} := {(0, 0, 1)};
α = (0, 1) : Φ_{α}(Lp(g_{5})) = 2, Φ_{α}(g_{5}) = 2T + 4;
Z := {1}, Z_{2} := Z_{2} ∪ {(0, 1, −2)};
α = (0, 2) : Φ_{α}(Lp(g_{5})) = 2, Φ_{α}(g_{5}) = 2T + 4;
Z := {−2}, Z_{2} := Z_{2} ∪ {(0, 2, −2)};
α = (1, 0) : Φ_{α}(Lp(g_{5})) = 0, Φ_{α}(Lp(g_{6})) = −1, Φ_{α}(g_{6}) = −T + 3;
Z := {3}, Z_{2} := Z_{2} ∪ {(1, 0, 3)};
α = (1, 1) : Φ_{α}(Lp(g_{5})) = Φ_{α}(Lp(g_{6})) = 0; Φ_{α}(Lp(g_{7})) = −1,
Φ_{α}(g_{7}) = −T^{2} + 4T − 3;
Z := {1, 3}, Z_{2} := Z_{2} ∪ {(1, 1, 3), (1, 1, 1)};
α = (2, 0) : Φ_{α}(Lp(g_{5})) = Φ_{α}(Lp(g_{6})) = Φ_{α}(Lp(g_{7})) = 0; Φ_{α}(Lp(g_{8})) = 1;
Φ_{α}(g_{8}) = T^{3} − 3T^{2} − T + 3;
Z := {0, 1, 2}, Z_{2} := Z_{2} ∪ {(2, 0, 0), (2, 0, 1), (2, 0, 2)}.

Remark 39.3.4. Cerlienco–Mureddu Correspondence and Algorithm (Chapter 33) can give some hint in the structure of Gianni–Kalkbrener Algorithm. Our informal discussion assumes that J is radical but holds in general. Gianni– Kalkbrener Algorithm, for any root α := (a_{1}, …, a_{i−1}) ∈ Z_{i−1}, considers the first Gröbner basis element

h = Lp(h)Z_{i}^{δ+1} + ⋯ ∈ K[Z_{1}, …, Z_{i}] \ K[Z_{1}, …, Z_{i−1}]

whose leading polynomial Lp(h) does not vanish at α.

In the same mood, Cerlienco–Mureddu considers a new point

x := (a_{1}, …, a_{i−1}, b) ∉ Z(J_{i})

and the first Gröbner basis element which does not vanish at it. Clearly both algorithms are choosing the same polynomial: in fact

δ + 1 = # {y ∈ Z(J_{i}) : π_{i−1}(y) = α = π_{i−1}(x)} = d.

Then

• Φ_{α}(h) | Φ_{α}(g), for each g ∈ G_{i} \ G_{iδ+1} because

g(y) = 0, for each y ∈ Z(J_{i}) : π_{i−1}(y) = α,

• Lp(g)(α) = 0 for each g ∈ G_{iδ}, because there is y ∈ Z(J_{i}) such that

π_{i−1}(y) = α = π_{i−1}(x)

whence g(y) = 0 and 0 = Lp(g)(π_{i−1}(y)) = Lp(g)(α).

In order to conclude our argument, we need to dispose of the elements g ∈ G_{iδ+1} \ G_{iδ} id est of the Gröbner basis element

g = Lp(g)Z_{i}^{δ+1} + ⋯;

to do so, we have just to remark that

• g(α) and h(α) are associate if T(g) ≻ T(h), because

g(y) = 0, for each y ∈ Z(J_{i}) : π_{i−1}(y) = α,

• and Lp(g)(α) = 0 if T(g) ≺ T(h), because there is y ∈ Z(J_{i}) such that

π_{i−1}(y) = α = π_{i−1}(x)

whence g(y) = 0 and 0 = Lp(g)(π_{i−1}(y)) = Lp(g)(α).

39.4 An Ecumenic Notion of Solving

As the decomposition algorithms (Chapter 35) were reducing primary decomposition of multivariate ideals to factorization of univariate polynomials, Trinks’ Algorithm (as most other solving algorithms) reduces multivariate zero-dimensional ideal solving, to univariate polynomial solving.

Most of these algorithms are ’ecumenic’, in the sense that they can be applied to any computational model of L which allows, in endlichvielen Schritten,

• to computationally perform the four operations in L, • and, for each univariate polynomial p(T) ∈ L[T], to ’solve’ it, id est to

produce the set {a ∈ L : p(a) = 0} of all the roots of p living in L,

and use these tools, given F, to ’solve’ the zero-dimensional ideal J generated by F, id est to produce the set Z(J)∩L^{r} of all the roots of J with coordinates in L.

Trink’s Algorithm (Figure 39.2) is a perfect instance of such ‘ecumenic’ algorithms: for instance, setting L := ℝ, it can be verbatim applied to a numerical analysis solver^{5}, or adapted in order to make use of Sturm Representation and Thom Codification of Algebraic Reals^{6}.

In the same way, Trink’s Algorithm can be easily adapted in order to make use of Kronecker’s (and Duval’s) Model; obviously the resulting algorithm (Figure 39.3) is a verbatim reformulation of the Zero-dimensional Prime Decomposition Algorithm discussed in Section 35.2. Such strict relation between ’solving’ and decomposing, which was already stressed in Section 34.5, is just a simple consequence of Kronecker–Duval Philosophy.

All over this Part we will preserve this ‘ecumenic’ approach to the notion of ’solving’, as much as the persented solvers will allow to do so; naturally, the most strict solver presented here is an integralist version of Kronecker–Duval Phylosophy.

39.5 *Delassus–Gunther Solver

Historical Remark 39.5.1. Trinks’ paper, dated 1978, is the first published application of Gröbner bases, except Buchberger’s thesis and paper. His result is an efficient adaptation and improvement of the proof of Hilbert’s Nullstellensatz given by Gröbner (Section 20.3).

Gröbner’s argument, when restricted to the zero-dimensional case, essentially computes iteratively the roots of each elimination ideal J_{i} by

• producing (via a suitable generic change of coordinates) a polynomial

f_{i}(Z_{1}, …, Z_{i}) := cZ_{i}^{d_{i}} + ∑_{j=0}^{d_{i}−1} h_{j}(Z_{1}, …, Z_{i−1})Z_{i}^{j} ∈ J_{i}, c ≠ 0,

• solving the univariate polynomials

f_{i}(a_{1}, …, a_{i−1}, Z_{i}) ∈ K[Z_{i}] for each (a_{1}, …, a_{i−1}) ∈ Z(J_{i−1})

5 Of course, such a statement must be taken cum grano salis: it forgets the illconditioning problem, which requires at least some suitable pre-processing before applying the Algorithm.

6 Chapter 13 dicusses both such representations and the techniques needed in order to solve the required polynomials

p(Z_{i}) := h(a_{1}, …, a_{i−1}, Z_{i}), h ∈ ℚ[Z_{1}, …, Z_{i}],

where each a_{i} ∈ ℝ is given by such representation.

Fig. 39.3. Trinks’ Algorithm in Kronecker’s Model

Z := Solve(F) where

F := (f_{1}, …, f_{u}) ⊂ Q := K[Z_{1}, …, Z_{r}], J ⊂ Q is the zero-dimensional ideal generated by F, Z := {(f_{1}, 𝔎_{1}, α_{1}), …, (f_{s}, 𝔎_{s}, α_{s})} where

f_{j} = (f_{j1}, …, f_{jr}) is an admissible sequence (Definition 8.2.2) in K[Z_{1}, …, Z_{r}], 𝔎_{j} := K[Z_{1}, …, Z_{r}]/(f_{j1}, …, f_{jr}), K ⊂ 𝔎_{j} ⊂ K is the finite extension explicitly given by f_{j}, α_{j} ∈ 𝔎_{j}^{r}

Z(J) := {α_{1}, …, α_{s}} ⊂ K^{r} Compute the reduced lexicographical Gröbner basis G of (f_{1}, …, f_{u}); Sort G := {g_{1}, …, g_{v}} by increasing maximal terms. Let g_{1} = ∏_{j=1}^{σ} q_{j}^{e_{j}} be the factorization of g_{1} over K, For j = 1 … σ let

f_{j} := (q_{j}), 𝔎_{j} := K[Z_{1}]/q_{j}, π_{j} : K[Z_{1}] → 𝔎_{j} be the canonical projection, α_{j} := π_{j}(Z_{1}) ∈ 𝔎_{j},

Z_{1} := {(f_{1}, 𝔎_{1}, α_{1}), …, (f_{σ}, 𝔎_{σ}, α_{σ})} For i = 2 … r do

Z_{i} := ∅, g := min(g ∈ G_{i} \ G_{i−1}) For each (f, 𝔎, α) ∈ Z_{i−1}, f = (f_{1}, …, f_{i−1}) , α = (a_{1}, …, a_{i−1}) do

h := g,

While Lp(h)(a_{1}, …, a_{i−1}) = 0 do h := Next(h, G)
p(Z_{i}) := h(a_{1}, …, a_{i−1}, Z_{i})
Let p = ∏_{j=1}^{σ} q_{j}^{e_{j}} be the factorization of p over 𝔎,
For j = 2 … σ let
f_{j} := (f_{1}, …, f_{i−1}, q_{j}),
𝔎_{j} := 𝔎[Z_{i}]/q_{j} ≅ K[Z_{1}, …, Z_{i}]/(f_{1}, …, f_{i−1}, q_{j})
π_{j} : 𝔎[Z_{i}] → 𝔎_{j} be the canonical projection,
α_{j} := (a_{1}, …, a_{i−1}, π_{j}(Z_{i})) ∈ 𝔎_{j}^{i},
Z_{i} := Z_{i} ∪ {(f_{1}, 𝔎_{1}, α_{1}), …, (f_{σ}, 𝔎_{σ}, α_{σ})}

Z := Z_{r}

• and including in Z(J_{i}) those roots (a_{1}, …, a_{i−1}, b) which are annihilating not only f_{i} but also a given basis of J_{i}.

Trinks proposal makes effective the ability of producing the required basis of each J_{i} and allows to produce univariate polynomials to be solved without performing changes of coordinates.

Gröbner’s argument, in turn, was an adapatation of the argument and solver by Kronecker (Section 20.4) which, in the zero-dimensional case^{7}, again consists into

• producing (via a suitable generic change of coordinates) polynomials

f_{i}(Z_{1}, …, Z_{i}) := cZ_{i}^{d_{i}} + ∑_{j=0}^{d_{i}−1} h_{j}(Z_{1}, …, Z_{i−1})Z_{i}^{j} ∈ _{i}, c ≠ 0,

• solving the univariate polynomials

f_{i}(a_{1}, …, a_{i−1}, Z_{i}) ∈ K[Z_{i}] for each (a_{1}, …, a_{i−1}) ∈ Z(I_{i−1})

• and including in Z(I_{i}) those roots (a_{1}, …, a_{i−1}, b) which are annihilating not only f_{i} but also a given basis of I_{i},

where each ideal I_{i−1} is obtained from I_{i} (I_{n} := J) via a suitable resultant computation.

Resultant is instead just a theoretical tool used in proving an interesting solver which anticipates some ideas by Macaulay: the original version^{8} was proposed by Delassus in 1987 but was flawed by the wrong assumption that the generic initial ideal (Definition 37.1.5) of a homogeneous ideal w.r.t. the lex ordering ≺ induced by X_{n} ≺ … ≺ X_{1} consists of the last terms w.r.t. ≺ while it is just Borel (Definition 37.2.7, Corollary 37.2.8); the flaw was found

(by Gunther and Robinson^{9}) and fixed (by Gunther^{10}) in 1913.  �

7 Unlike Gröbner’s argument which was not intended as an effictive solver and was

turn into such by Trinks, Kronecker’s argument was an effective solver.

The restriction to the zero-dimensional case is done here to simplify the argument but is not required by Kronecker’s solver which in fact applies also to non-unmixed ideals.

The details on the general case are discussed in Sections 20.3 and 20.4. 8 Delassus E., Sur les systèmes algébriques et leurs relations avec certains systèmes

d´’equations aux dérivées partielles. Ann. Éc. Norm. 3^{e} série 14 (1897) 21–44 9 Gunther, N. Sur les caractéristiques des systémes d’equations aux dérivées par-

tialles, C.R. Acad. Sci. Paris 156 (1913), 1147–1150 and Robinson, L.B. Sur les systémes d’équations aux dérivées partialles C.R. Acad. Sci. Paris 157 (1913), 106–108

10 Gunther, N. Sur la forme canonique des systèmes d’équations homogènes (in russian) [Journal de l’Institut des Ponts et Chaussées de Russie] Izdanie Inst. Inz̆. Putej Soobs̆c̆enija Imp. Al. I. 84 (1913) and Gunther, N. Sur la forme canonique des équations algébriques, C.R. Acad. Sci. Paris 157 (1913), 577–80 .

In order to present the solver proposed by Delassus–Gunther we must slightly adapt the notation used; we assume J ⊂ Q = K[Z_{1}, …, Z_{r}] to be homogeneous and we denote, for each d ∈ ℕ,

W_{d} := {τ ∈ W, deg(τ) = d} and^{11} J_{d} := J ∩ Span_{K}(W_{d}).

We assume to have performed a generic change of coordinates and we consider the (deg)-revlex ordering < induced by Z_{1} < Z_{2} < … < Z_{r} and for each (homogeneous) polynomial f = ∑_{t∈W} c(f, t)t ∈ Q we denote

L_{<}(f) := min_{<}(t ∈ W : c(f, t) ≠ 0)

and, for each (homogeneous) set F ⊂ Q,

L_{<}{F } := {L_{<}(f) : f ∈ F } and L_{<}(F) := {τL_{<}(f) : f ∈ F, τ ∈ W}.

Remark 39.5.2.

(1) if ≺ denotes the (deg)-lex ordering induced by Z_{1} ≻ Z_{2} ≻ … ≻ Z_{r} we have T_{≺}(f) = L_{<}(f) for each (homogeneous) polynomial f ∈ Q and T_{≺}(F) = L_{<}(F) for each (homogeneous) set F ⊂ Q;

(2) for the homogeneous ideal J and each d ∈ ℕ, we have

L_{<}(J)_{d} = L_{<}{J}_{d} = L_{<}{J_{d}};

(3) there is a (minimal) value^{12} D ∈ ℕ which satisfies, for each d ∈ ℕ,

L(Span_{K}{ωf : ω ∈ W_{d}, f ∈ J_{D}}) = {ωL(f) : ω ∈ W_{d}, f ∈ J_{D}};

(4) each set L_{<}{J_{d}}, d ≥ D, satisfies^{13} for each ℓ, ℓ′, 1 ≤ ℓ < ℓ′ ≤ r,

Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ L_{<}{J_{d}} ⟹ Z_{1}^{a_{1}} ⋯ Z_{ℓ}^{a_{ℓ}+1} ⋯ Z_{ℓ′}^{a_{ℓ′} −1} ⋯ Z_{r}^{a_{r}} ∈ L_{<}{J_{d}};

11 the notation J_{d} denotes here the set of the homogeneous members of J of degree d and must not be indentify with the previous notation where J_{i} denotes the members of J depending only on the first j variables.

12 We can set D := max{deg(g) : g ∈ G} where G is a Gröbner basis of J wrt ≺ but the existence can be easily derived (as for the finiteness of Gröbner bases) by Hilbert’s Nullstellensatz and this is the approach used by Gunther.

13 This is a direct consequence of Corollary 37.2.8 applied to the set T_{≺}(f) = L_{<}(f)

and to the (deg)-lex ordering ≺ induced by Z_{r} ≺ … ≺ Z_{2} ≺ Z_{1}.

Delassus’ mistake is to assume that T_{≺}(J_{d}) = L_{<}(J_{d}) is the set L(d) consists of the first #L_{<}(J_{d}) terms w.r.t. the (deg)-revlex < induced by Z_{1} < Z_{2} < … < Z_{r} which tantamount to the last #T_{≺}(J_{d}) terms w.r.t. the (deg)-lex ordering ≺ induced by Z_{r} ≺ … ≺ Z_{2} ≺ Z_{1}.

In its Lemma (cf. Section 23.3) Macaulay was considering the same set L(d), #L(d) = #L_{<}(J_{d}) as Delassus and presented it, as Delassus, in terms of the (deg)-revlex ordering < and not in terms of the (deg)-lex ordering ≺.

(5) Denoting, for each d, N(J_{d}) := W_{d} \ L_{<}{J_{d}}, for each τ ∈ L_{<}{J_{d}}, we have Can(τ, J, ≺) ∈ Span_{K}(N(J_{d})) and we can set

g_{τ} := τ − Can(τ, J, ≺) ∈ Span_{K}(N(J_{d})) and G_{d} := {g_{τ} : τ ∈ L_{<}{J_{d}}}

having τ = L_{<}(g_{τ}) = T_{≺}(g_{τ}) and L_{<}{G_{d}} = L_{<}{J_{d}}.

(6) Z(J) = {(a_{1}, …, a_{r}) ∈ K^{r} : g(a_{1}, …, a_{r}) = 0 for each g ∈ G_{D}}.  �

Theorem 39.5.3 (Delassus–Gunther). With the present notation and assumptions, let γ_{1}, …, γ_{r} ∈ ℕ, ∑_{i=1}^{r} γ_{i} = D, be the values such that

Ω := Z_{1}^{γ_{1}} ⋯ Z_{r−1}^{γ_{r−1}}Z_{r}^{γ_{r}} = max_{<}(L_{<}{J_{D}}) = max_{<}(L_{<}{G_{D}}) = min_{≺}(T_{≺}{J_{D}}).

Then

(1) if γ_{1} = 0 then gcd(G_{D}) = 1;

(2) if γ_{1} ≠ 0 then (a) h := gcd(G_{D}) ≠ 1;

(b) L_{<}(h) = Z_{1}^{γ_{1}};

(c) for each d ∈ ℕ it holds

L(Span_{K}{ωg : ω ∈ W_{d}, g ∈ G_{D}}) = {ωL(g) : ω ∈ W_{d}, g ∈ G_{D}};

(3) max_{<}{L_{<}(g/h) : g ∈ G_{D}} = Z_{2}^{γ_{2}} ⋯ Z_{r−1}^{γ_{r−1}}Z_{r}^{γ_{r}};

(4) if γ_{1} = 0 the set H := G_{D} ∩ K[Z_{2}, …, Z_{r}] satisfies, for each d ∈ ℕ,

L(Span_{K}{ωf : ω ∈ U_{d}, f ∈ H}) = {ωL(f) : ω ∈ U_{d}, f ∈ H},

where we have set U_{d} := W_{d} ∩ K[Z_{2}, …, Z_{r}];

(5) if γ_{1} = 0 and

(a_{2}, …, a_{r}) ∈ Z(H) := {(a_{2}, …, a_{r}) ∈ K^{r−1} : g(a_{2}, …, a_{r}) = 0, g ∈ H},

then 1 ≠ h(Z_{1}, a_{2}, …, a_{r}) = gcd(g(Z_{1}, a_{2}, …, a_{r}) : g ∈ G_{D}) ∈ K[Z_{1}];

(6) moreover h(Z_{1}, a_{2}, …, a_{r}) = (Z_{1} − a_{1})^{deg(g)} for some a_{1} ∈ K.

Proof.

(1) Remark that the first element, w.r.t. < in L_{<}{J_{D}} is Z_{1}^{D} and that

Z_{1}^{D} − Can(Z_{1}^{D}, J, ≺) ∈ G_{D}

so that

h := gcd(G_{D}) ≠ 1 ⟹ gcd(G_{D}) ∈ K[Z_{2}, … Z_{r}][Z_{1}] \ K[Z_{2}, … Z_{r}].

If γ_{1} = 0, Ω ∈ K[Z_{2}, … Z_{r}] and g_{Ω} ∈ K[Z_{2}, … Z_{r}]^{14} so that

h = gcd(G_{D} ∈ K[Z_{2}, … Z_{r}]).

Thus gcd(G_{D}) = 1.

14 As a consequence of the elimination property of the lex ordering ≺ which is explicitly stated by Gunther.

(2) Assume now that γ_{1} ≠ 0.

(a) Let us consider variables U_{τ} , W_{τ} , τ ∈ L_{<}{J_{D}}, and the polynomials

f := ∑_{τ∈L_{<}{J_{D}}} U_{τ} g_{τ} , g := ∑_{τ∈L_{<}{J_{D}}} W_{τ} g_{τ} ∈ K[U_{τ} , W_{τ} , Z_{2}, … Z_{r}][Z_{1}];

Sylvester resultant grants (Proposition 6.6.7) the existence of polynomials p, q ∈ K[U_{τ} , W_{τ} , Z_{2}, … Z_{r}][Z_{1}] such that

Res(f, g) := pf + qg ∈ K[U_{τ} , W_{τ} , Z_{2}, … Z_{r}];

moreover Res(f, g) is necessarily linear and homogeneous in terms of members of the set F := {ωg_{τ} , τ ∈ L_{<}{J_{D}}, ω ∈ W_{d}} where d := deg(p) = deg(q). By Remarks 39.5.2.(3) and (5),

ΩZ_{r}^{d} := Z_{1}^{γ_{1}} ⋯ Z_{r−1}^{γ_{r−1}}Z_{r}^{γ_{r}+d} = max_{<}(L_{<}{F }).

As a consequence, since γ_{1} ≠ 0, there are elements f′ ∈ F, for instance f′ = Z_{r}^{d}g_{Ω}, for which

L_{<}(f′) ∈ K[U_{τ} , W_{τ} , Z_{2}, … Z_{r}][Z_{1}] \ K[U_{τ} , W_{τ} , Z_{2}, … Z_{r}]

thus getting a contradiction unless Res(f, g) = 0 and f, g have a common factor in K[U_{τ} , W_{τ} , Z_{2}, … Z_{r}][Z_{1}]. Such factor is necessarily a member of K[Z_{2}, … Z_{r}][Z_{1}], thus proving that h := gcd(G_{D}) ≠ 1.

(b) We necessarily have L_{<}(h) = Z_{1}^{γ} for some γ ∈ ℕ. Also

max_{<}{L_{<}(g/h) : g ∈ G_{D}} = Z_{1}^{γ_{1}−γ}Z_{2}^{γ_{2}} ⋯ Z_{r−1}^{γ_{r−1}}Z_{r}^{γ_{r}}

and, setting F′ := {ωg/h : g ∈ G_{D}, ω ∈ W_{d}}, we have (again by Remarks 39.5.2.(3) and (5))

max_{<}{L_{<}(f : f ∈ F′) = Z_{1}^{γ_{1}−γ}Z_{2}^{γ_{2}} ⋯ Z_{r−1}^{γ_{r−1}}Z_{r}^{γ_{r}+d}.

Since clearly gcd(F′) = 1 we conclude by (1) and (2.a) above that γ_{1} − γ = 0 id est γ_{1} = γ.

(c) is trivial.

(3) is a trivial consequence of (2).

(4) If γ_{1} = 0, then H := G_{D} ∩ K[Z_{2}, …, Z_{r}] ≠ ∅ and^{15}

H = {g_{τ} ∈ G_{D} : τ ∈ W ∩ K[Z_{2}, … Z_{r}]}.

The claim then is a direct application of Remark 39.5.2.(3).

15 Again a consequence of the elimination property of the lex ordering ≺ explicitly stated by Gunther.

(5) Let us again consider the resultant

Res(f, g) := pf + qg ∈ K[U_{τ} , W_{τ} , Z_{2}, … Z_{r}]

of f := ∑_{τ∈L_{<}{J_{D}}} U_{τ} g_{τ} and g := ∑_{τ∈L_{<}{J_{D}}} W_{τ} g_{τ}; since gcd(G_{D}) = 1 we have gcd(f, g) = 1 and Res(f, g) ≠ 0. Denoting V the set of the terms in the variables {U_{τ} , W_{τ} , τ ∈ L_{<}{J_{D}}} we therefore have Res(f, g) = ∑_{υ∈V} c_{υ}υ. Each c_{υ} depends linearly on the elements in

F := {ωg_{τ} , τ ∈ L_{<}{J_{D}}, ω ∈ W_{d}},

where d := deg(p) = deg(q), and is independent on Z_{1}: c_{υ} ∈ K[Z_{2}, … Z_{r}]. Therefore each c_{υ} depends linearly on the elements in

F′ := {ωg, g ∈ H, ω ∈ W_{d} ∩ K[Z_{2}, … Z_{r}]}

and we have c_{υ}(a_{2}, …, a_{r}) = 0, Res(f, g)(U_{τ} , W_{τ} , a_{2}, …, a_{r}) = 0 for each (a_{2}, …, a_{r}) ∈ Z(H), so that

gcd (f(U_{τ} , W_{τ} , a_{2}, … a_{r}, Z_{1}), g(U_{τ} , W_{τ} , a_{2}, … a_{r}, Z_{1})) ≠ 1

id est 1 ≠ h(Z_{1}, a_{2}, …, a_{r}) := gcd(g(Z_{1}, a_{2}, …, a_{r}) : g ∈ G_{D}) ∈ K[Z_{1}].

(6) Since this is Gröbner’s Allgemeine Nulldimensional Basissatz (Theorem 34.2.4) I can skip the interesting, but not immediate, proof proposed

by Gunther^{16}.  �

16 It begins by considering a substitution

Z_{1} = U_{1}, Z_{2} = AU_{1} + U_{2}, Z_{3} = U_{3}, …, Z_{r} = U_{r},

where A is a variable, and the corresponding equations in K[A][U_{1}, …, U_{r}] which sont de fonction holomorphes de A dans la voisinage de A = 0.

28 39. Trinks

40. Stetter

The crucial improvement by Gianni and Kalkbener of Trinks’ Algorithm is dated 1987; the next year, Auzinger and Stetter proposed an alternative algorithm for solving a radical 0-dimensional ideal which was later generalized by Stetter and Möller to the general setting; the original proposal made no reference to Gröbner techniques^{1} being based on Numerical Analysis techniques: given a zero dimensional ideal J ⊂ Q, Auzinger–Stetter’s Theorem states that, for each f ∈ A := Q/J, the linear form Φ_{f} : A → A describing the multiplication by f in A has the evaluation of f at the roots of J as its eigenvalues with the proper multiplicity; moreover, if we fix a K-basis of A b = {b_{1}, …, b_{s}} and we denote A_{f} the matrix representing Φ_{f} w.r.t. such basis, then (b_{1}(α), …, b_{s}(α))^{T} is an eigenvector for f(α) for each α ∈ Z(J).

Thus, provided that A_{f} is non-derogatory, id est its Jordan form has a single Jordan block associated with each eigenvalue, it is sufficient to choose as b a basis which includes the linear basis of the subspace of A consisting of all its linear forms and use linear dependency to express each variable Z_{i} in terms of such linear basis.

After setting the proper notation (Section 40.1) we present Auzinger– Stetter’s Theorem (Section 40.2) and discuss how to apply it for solving a 0-dimensional radical ideal (Section 40.3). The extension to the general case being based on duality, I preliminarily discuss the relation between duality and Auzinger–Stetter’s technique (Section 40.4) before presenting Möller– Stetter’s extension of Auzinger–Stetter’s Theorem (Section 40.5).

After specializing this result to the univariate case, thus obtaining the expected statement (Section 40.6) and discussing derogatoriness (Section 40.7 and 40.10), I finally present how Stetter’s Algorithm can be performed using Gröbner basis techniques (Section 40.8 and 40.9).

1 While, the presentation here is centered around the notion of Gröbner representation, one must remark that such notion was introduced later in order to provide a convenient frame to present Auzinger–Stetter’s Algorithm.

40.1 Endomorphisms of an Algebra

Let Q := K[Z_{1}, …, Z_{r}], W its monomial K-basis and K the algebraic closure of K. In order to simplify the notation let us wlog assume K = K to be algebraically closed.

Let J ⊂ Q be a zero-dimensional ideal, deg(J) = s, and A := Q/J the corresponding quotient algebra, which satisfies dim_{K}(A) = s.

For any f ∈ Q, we will denote [f] ∈ A its residue class modulo J and Φ_{f} the endomorphism Φ_{f} : A → A defined by

Φ_{f}([g]) = [fg] for each [g] ∈ A.

Clearly Φ_{f} = Φ_{h} iff [f] = [h].

If we fix any K-basis b = {[b_{1}], …, [b_{s}]} of A so that A = Span_{K}(b), then for each g ∈ Q, there is a unique (row) vector, the Gröbner description of g (Definition 29.3.3),

Rep(g, b) := (γ(g, b_{1}, b), …, γ(g, b_{s}, b)) ∈ K^{s}

which satisfies

[g] = ∑_{j} γ(g, b_{j}, b)[b_{j}]

and the endomorphism Φ_{f} is naturally represented by the square matrix

M([f], b) = (γ(fb_{i}, b_{j}, b)) : Φ_{f}(b_{i}) = [fb_{i}] = ∑_{j} γ(fb_{i}, b_{j}, b)[b_{j}].

Recall that a Gröbner representation (Definition 29.3.3) of J is the assignement of

• a K-basis b = {[b_{1}], …, [b_{s}]} ⊂ A and

• the square matrices A_{h} := (a_{ij}^{(h)}) = M([Z_{h}], b) for each h, 1 ≤ h ≤ s,

and that a Gröbner representation is called a linear representation (Definition 29.3.3) iff b = {[1], [τ_{2}], …, [τ_{s}]} = N_{<}(J) wrt a term ordering < and remark that, for each f(Z_{1}, …, Z_{r}) ∈ Q, M([f], b) = f(A_{1}, …, A_{r}).

An alternative way of representing a zero-dimensional ideal J ⊂ Q and the related quotient algebra A is via its dual space (Section 28.1)

𝔏(J) := {ℓ ∈ Q^{∗} : ℓ(g) = 0 for each g ∈ J} ⊂ Q^{∗}

where Q^{∗} := Hom_{K}(Q, K) is the K-vectorspace consisting of all K-linear functionals ℓ : Q → K.

Clearly we have dim_{K}(𝔏(J)) = s and to each K-basis 𝕃 := {λ_{1}, ⋯, λ_{s}} of 𝔏(J) is associated a Lagrange K-basis q = {[q_{1}], …, [q_{s}]} which is biorthogo-

nal to 𝕃 id est λ_{i}(q_{j}) = δ_{ij} = \{ \begin{matrix}1 & if i = j \\ 0 & if i ≠ j. \end{matrix}

In particular, since, for each i, j, h,

λ_{j}(Z_{h}q_{i}) = λ_{j} (∑_{l} a_{il}^{(h)}q_{l}) = ∑_{l} a_{il}^{(h)}λ_{j}(q_{l}) = a_{ij}^{(h)},

to each basis 𝕃 := {λ_{1}, ⋯, λ_{s}} of 𝔏(J) is associated the Gröbner representation

• q = {[q_{1}], …, [q_{s}]} ⊂ A : λ_{i}(q_{j}) = δ_{ij} for each i, j,

• Q_{h} := (λ_{j}(Z_{h}q_{i}))_{ij}.

Example 40.1.1. Set Q := ℂ[Z_{1}, Z_{2}] and

J := (Z_{1}^{3} − Z_{1}^{2}, Z_{1}Z_{2} − Z_{1} − Z_{2} + 1, Z_{2}^{3} + Z_{1}^{2} − 1)

which is a Gröbner basis wrt the degree compatible ordering induced by Z_{1} < Z_{2}.

As a consequence we can choose as Gröbner representation the linear representation b = {1, Z_{1}, Z_{2}, Z_{1}^{2}, Z_{2}^{2}},

A_{1} = ( \begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ −1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ −1 & 1 & 0 & 0 & 1 \end{matrix} ) , A_{2} = ( \begin{matrix}0 & 0 & 1 & 0 & 0 \\ −1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ −1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & −1 & 0 \end{matrix} ) .

It is easy to verify that 𝔏(J) = Span_{K}(𝕃), 𝕃 := {λ_{1}, …, λ_{5}} with

\begin{matrix}λ_{1}(p) = p(0, 1), & λ_{2}(p) = \frac{∂p}{∂Z_{1}} (0, 1), \\ λ_{3}(p) = p(1, 0), & λ_{4}(p) = \frac{∂p}{∂Z_{2}} (1, 0), & λ_{5}(p) = \frac{∂^{2}p}{2∂Z_{2}^{2}} (1, 0) \end{matrix}

whose associated Gröbner representation is q = {q_{1}, q_{2}, q_{3}, q_{4}, q_{5}} with

\begin{matrix}q_{1} = −Z_{1}^{2} + 1, & q_{2} = −Z_{1}^{2} + Z_{1}, \\ q_{3} = Z_{1}^{2}, & q_{4} = Z_{1}^{2} + Z_{2} − 1, & q_{5} = Z_{1}^{2} + Z_{2}^{2} − 1, \end{matrix}

Q_{1} = ( \begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ) , Q_{2} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) .

�

Between the two bases b and q there are the basis transformations

M_{bq} := (γ(b_{i}, q_{j}, q)) and M_{qb} := (γ(q_{i}, b_{j}, b))

so that, for each i,

[b_{i}] = ∑_{j} γ(b_{i}, q_{j}, q)[q_{j}] and [q_{i}] = ∑_{j} γ(q_{i}, b_{j}, b)[b_{j}];

naturally, we have M_{bq} = M_{qb}^{−1}, and

M([f], b) = M_{bq}M([f], q)M_{qb} = M_{bq}M([f], q)M_{bq}^{−1}

so that M([f], q) and M([f], b) are similar and share the same eigenvalues and Jordan normal form.

Example 40.1.2. Continuing Example 40.1.1 we have

M_{bq} = ( \begin{matrix}1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{matrix} ) , M_{qb} = ( \begin{matrix}1 & 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ −1 & 0 & 1 & 1 & 0 \\ −1 & 0 & 0 & 1 & 1 \end{matrix} ) ,

and it is easy to check the relations M_{bq} = M_{qb}^{−1}, M_{bq}Q_{1} = A_{1}M_{bq}, M_{bq}Q_{2} = A_{2}M_{bq}. Setting

J_{1} = ( \begin{matrix}0 & 1 \\ & 0 \\ & & 1 \\ & & & 1 \\ & & & & 1 \end{matrix} ) , J_{2} = ( \begin{matrix}1 \\ & 1 \\ & & 0 & 1 \\ & & & 0 & 1 \\ & & & & 0 \end{matrix} );

we have, for i = 1, 2, M_{bq}J_{i} = A_{i}M_{bq}, and hence J_{i} = Q_{i}.

From M_{bq}J_{i} = A_{i}M_{bq}, we obtain M_{qb}A_{i} = J_{i}M_{qb} and A_{i}^{T} M_{qb}^{T} = M_{qb}^{T}J_{i}^{T} = M_{qb}^{T}Q_{i}^{T} , thus allowing to easily deduce eigenvalues and Jordan normal forms also for A_{i}^{T}.

In fact we have M̌_{qb}J̌_{i} = A_{i}^{T} M̌_{qb} with M̌_{qb} = ( \begin{matrix}−1 & −1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & −1 & −1 \\ 1 & 0 & 0 & 0 & 0 \end{matrix} ) ,

J̌_{1} = ( \begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & 0 & 1 \\ & & & & 0 \end{matrix} ) and J̌_{2} = ( \begin{matrix}0 & 1 \\ & 0 & 1 \\ & & 0 \\ & & & 1 \\ & & & & 1 \end{matrix} ) .

� Remark 40.1.3. Denote P the s-square backward identity matrix

P := (p_{ij}), p_{ij} = \{ \begin{matrix}1 & if i + j = s + 1 \\ 0 & if i + j ≠ s + 1 \end{matrix}

which satisfy P^{−1} = P.

From a relation MJ = AM where A is an s-square matrix, M is invertible, N := M^{−1}, and J is the Jordan matrix of A, we obtain JN = NA and

A^{T} (N^{T} P) = N^{T} J^{T} P = (N^{T} P)(P J^{T} P)

so that P J^{T} P is the Jordan matrix of A^{T} — whose eigenvalues thus are the same (with the same multiplicity) as the ones of A — and Ň := N^{T} P its eigenspace matrix.

Given a square matrix N = (n_{ij}), the matrix Ň = N^{T} P = \binom{}{ň_{ij}} can be obtained from N by, equivalently, either

• writing from the right to the left the columns of N^{T},

• clock-right \frac{π}{4} -rotating N or

• setting ň_{ij} = n_{s−j i}, for each 1 ≤ i, j ≤ s.  �

40.2 Toward Auzinger–Stetter’s Theorem

With the same notation as in the previous section let us fix

• a Gröbner representation

b = {[b_{1}], …, [b_{s}]} ⊂ A, A_{h} := (a_{ij}^{(h)}) = M([Z_{h}], b), 1 ≤ h ≤ r;

• a basis 𝕃 := {λ_{1}, ⋯, λ_{s}} of 𝔏(J);

• the conjugate Gröbner representation

q = {[q_{1}], …, [q_{s}]} ⊂ A, Q_{h} := (λ_{j}(Z_{h}q_{i}))_{ij} ,

where q is the Lagrange basis satisfying λ_{i}(q_{j}) = δ_{ij} for each i, j,

and let us denote

• M_{bq} := (γ(b_{i}, q_{j}, q)) and M_{qb} := (γ(q_{i}, b_{j}, b)) the basis transformation matrices;

• M̌_{qb} the matrix obtained from M_{qb} according the construction described in Remark 40.1.3;

• J_{h} and J̌_{h}, 1 ≤ h ≤ r the Jordan normal form matrices for A_{h} and A_{h}^{T};

• for each f ∈ Q/J = A

A_{f} := M([f], b) = (γ(fb_{i}, b_{j}, b)) : Φ_{f}(b_{i}) = [fb_{i}] = ∑_{j} γ(fb_{i}, b_{j}, b)[b_{j}];

• J_{f} and J̌_{f} the Jordan normal form matrices for A_{f} and A_{f}^{T}.

Let us also consider the set

Z(J) := {α ∈ K^{r} : f(α) = 0 for each f ∈ J}.

Lemma 40.2.1 (Auzinger–Stetter). With the present notation it holds

γ(b_{i}, q_{j}, q) = λ_{j}(b_{i}), 1 ≤ i, j ≤ s.

Proof. For each f ∈ A, ∑_{j} γ(f, q_{j}, q)[q_{j}] = f = ∑_{j} λ_{j}(f)[q_{j}].

The first equality follows from the definition of γ, the second from the property of the Lagrange basis. The claim then follows by the linear inde-

pendency of q.  �

Corollary 40.2.2. Each i^{th} row of M_{bq} is the vector (λ_{1}(b_{i}), …, λ_{s}(b_{i})) of the evaluation of the basis element b_{i} at the functional basis 𝕃.

Each j^{th} column of M_{bq} is the vector (λ_{j}(b_{1}), …, λ_{j}(b_{s}))^{T} of the evalua-

tion of the basis b at the functional λ_{j}.  �

Lemma 40.2.3 (Auzinger–Stetter). For each α ∈ Z(J) the vector

(b_{1}(α), …, b_{s}(α))^{T}

is an eigenvector of the matrix A_{f} for the eigenvalue f(α).

Proof. For each i, 1 ≤ i ≤ s, we have [fb_{i}] = Φ_{f}([b_{i}]) = ∑_{j} γ(fb_{i}, b_{j}, b)[b_{j}] so

that f(α)b_{i}(α) = ∑_{j} γ(fb_{i}, b_{j}, b)b_{j}(α). Thus the claim follows trivially. �

Lemma 40.2.4 (Möller). The following holds:

(1) for any λ ∈ K, λ is an eigenvalue for Φ_{f} iff J : (f − λ) ≠ J;

(2) the corresponding eigenspace is the set {[h] : h ∈ J : (f − λ)};

(3) [h] = ∑_{i} β_{i}[b_{i}] ∈ J : (f − λ) iff (β_{1}, …, β_{s})^{T} is an eigenvector of A_{f}^{T} for λ.

Proof. For each [h] = ∑_{i} β_{i}[b_{i}] ∈ A we have

Φ_{f}([h]) = ∑_{j} β_{j}Φ_{f}([b_{j}])
= ∑_{j} β_{j} ∑_{i} γ(fb_{j}, b_{i}, b)[b_{i}]

= ∑_{i} (∑_{j} β_{j}γ(fb_{j}, b_{i}, b)) [b_{i}]

so that, for v := (β_{1}, …, β_{s})^{T} we have

λ[h] = Φ_{f}([h]) ⟺ λβ_{i} = ∑_{j} β_{j}γ(fb_{j}, b_{i}, b)∀i ⟺ λv = A^{T} v.

For any h ∉ J we have the obvious equivalences

λ[h] = Φ_{f}([h]) ⟺ λ[h] = [fh]
⟺ (λ − [f]) [h] = 0
⟺ (λ − f) h ∈ J
⟺ h ∈ J : (f − λ)
whence the claim.  �

Definition 40.2.5. A matrix is called non-derogatory if, equivalently,

all its eigenspaces have dimension 1; its Jordan form has a single Jordan block associated with each eigenvalue.

�

Theorem 40.2.6 (Auzinger–Stetter). The set {f(α) : α ∈ Z(J)} is the set of eigenvalues of A_{f}. If A_{f} is non-derogatory, each eigenspace of A_{f} for f(α) is spanned by (b_{1}(α), …, b_{s}(α))^{T}.

Proof. A direct consequence of Lemmata 40.2.3 and 40.2.4.  �

Corollary 40.2.7. The set {f(α) : α ∈ Z(J)} is the set of eigenvalues of A_{f}^{T}. If A_{f} is non-derogatory, such is also A_{f}^{T} and, for each i,

(1) the eigenspace of A_{f}^{T} for f(α_{i}) is spanned by

(γ(q_{i}, b_{1}, b), ⋯, γ(q_{i}, b_{s}, b)))^{T} .

(2) J : (f − f(α_{i})) = J + (q_{i}).

�

Example 40.2.8. Continuing Example 40.1.1, the eigenspace of A_{1} (respectively: A_{2}) for the eigenvalue 0 is spanned by (1, 0, 1, 0, 1)^{T} (respectively: (1, 1, 0, 1, 0)^{T}) while (1, 1, 0, 1, 0)^{T} (respectively: (1, 0, 1, 0, 1)^{T}) are just eigenvectors for the eigenvalue 1 whose eigenspace has dimension 3 (respectively: 2). The eigenspaces for 0 have dimension 1 for both A_{1} and A_{2}, those for 1 have dimension respectively 3 and 2; thus neither matrix is non-derogatory.

We also have:

• J : Z_{1} = (Z_{1} − Z_{1}^{2}) + J and (0, 1, 0, −1, 0)^{T} spans the eigenspace of A_{1}^{T} for 0;

• J : (Z_{1} − 1) = (Z_{1}^{2}, Z_{2} − 1) + J and the eigenspace of A_{1}^{T} for 1 is spanned by

{(−1, 0, 0, 1, 1)^{T} , (−1, 0, 1, 1, 0)^{T} , (0, 0, 0, 1, 0)^{T} }
= {(0, 0, −1, 0, 1)^{T} , (−1, 0, 1, 0, 0)^{T} , (0, 0, 0, 1, 0)^{T} };

• J : Z_{2} = (Z_{2}^{2} + Z_{1}^{2} − 1) + J and the eigenspace of A_{2}^{T} for 0 is spanned by {(−1, 0, 0, 1, 1)^{T} }.

• J : (Z_{2} − 1) = (Z_{1} − 1) + J and the eigenspace of A_{2}^{T} for 1 is spanned by

{(0, 1, 0, −1, 0)^{T} , (1, 0, 0, −1, 0)^{T} } = {(1, −1, 0, 0, 0)^{T} , (0, 1, 0, −1, 0)^{T} }. �

The relevant aspect of Auzinger–Stetter’s Theorem 40.2.6 is that while both eigenvalues and eigenvectors of A_{f} intrinsecally depend on the roots of J their actual values are precise functions of the choice of the matrix A_{f} and of the basis b; one can therefore expects that for a proper choice of f and b an eigenvalue computation can allow to deduce the roots of J.

40.3 Auzinger–Stetter: The Radical case

Let us preliminarly assume that J is radical and see whether the remark above leads to something.

The radicality assumption implies that J has s = deg(J) different roots in K^{r}:

Z(J) = {α_{1}, …, α_{s}} ⊂ K^{r}, α_{j} = (a_{1}^{(j)}, …, a_{r}^{(j)}).

Thus we can wlog identify each functional λ_{j} with the evaluation at the root α_{j}:

λ_{j} : Q → K, p(Z_{1}, …, Z_{r}) ↦ λ_{j}(p) = p(a_{1}^{(j)}, …, a_{r}^{(j)})

and q is the corresponding Lagrange basis.

A matrix A_{f} is non-derogratory if and only if f(α_{i}) ≠ f(α_{j}) for each i ≠ j. Clearly for a generic linear form Y = ∑_{h} c_{h}Z_{h}, A_{Y} is non-derogatory. Thus if we choose a linear form which separates Z(J) id est it satisfies the condition

(AS.1) Y = ∑_{h} c_{h}Z_{h} is such that β_{i} := ∑_{h} c_{h}a_{h}^{(i)} ≠ ∑_{h} c_{h}a_{h}^{(j)} =: β_{j} for each i ≠ j

then A_{Y} and A_{Y}^{T} are non-derogatory and have s distinct eigenvalues

β_{j} := ∑_{h} c_{h}a_{h}^{(j)}, 1 ≤ j ≤ s

whose associated eigenspaces are generated respectively by

(b_{1}(α_{j}), …, b_{s}(α_{j}))^{T} and (γ(q_{j}, b_{1}, b), ⋯, γ(q_{j}, b_{s}, b))^{T} .

In order to deduce the α_{j}s from these eigenvectors, the trick consists in a clever choice of the basis b. The efficient choice is the original one proposed by Auzinger–Stetter: let us denote V the K-vectorspace

V := Span_{K}{[1], [Z_{1}], … [Z_{r}]}

and let δ := dim_{K}(V) ≤ s; then, up to reenumerating the variables, we can wlog assume that

• V = Span_{K}{[1], [Z_{1}], … [Z_{δ−1}]}

• {[1], [Z_{1}], … [Z_{δ−1}]} is a K-basis of V,

• there are c_{il} ∈ K, 0 ≤ l < δ ≤ i ≤ r such that [Z_{i}] = c_{i0} + ∑_{l=1}^{δ−1} c_{il}[Z_{l}].

Moreover, the knowledge of the matrices A_{h} allows to deduce, by easy linear algebra, both δ and the c_{il}s.

We can therefore choose a basis b which satisfies the condition

(AS.2) b = ([b_{1}], …, [b_{s}]) is such that

b_{1} = 1, b_{i} = Z_{i−1}, 1 < i ≤ δ = dim_{K}(V)

so that

V := Span_{K}{[1], [Z_{1}], … [Z_{r}]} = Span_{K}{[1], [Z_{1}], … [Z_{δ−1}]}
= Span_{K}{[b_{1}], …, [b_{δ}]};

thus the eigenvectors corresponding to α_{j} = (a_{1}^{(j)}, …, a_{r}^{(j)}) are

(1, a_{1}^{(j)}, …, a_{δ−1}^{(j)}, b_{δ+1}(α_{j}), …, b_{s}(α_{j}))^{T}

and the other coordinates of α_{j} can be deduced from a_{i}^{(j)} = c_{i0}+∑_{l=1}^{δ−1} c_{il}a_{l}^{(j)}. In conclusion

Theorem 40.3.1 (Auzinger–Stetter). With the present notation and under the assumption that J is radical, then it holds

(1) each j^{th} column (b_{1}(α_{j}), …, b_{s}(α_{j}))^{T} of M_{bq} is an eigenvector of each A_{f}, f ∈ Q, for the eigenvalue f(α_{j});

(2) each j^{th} row (γ(q_{j}, b_{1}, b), ⋯, γ(q_{j}, b_{s}, b))^{T} of M_{qb} is an eigenvector of each A_{f}^{T} , f ∈ Q, for the eigenvalue f(α_{j});

(3) for each f ∈ Q, it holds (a) the eigenvalues of A_{f} and A_{f}^{T} are {f(α_{j}) : 1 ≤ j ≤ s};

(b) the eigenspace of A_{f} for λ ∈ K is

Span_{K}{(b_{1}(α_{j}), …, b_{s}(α_{j}))^{T} : f(α_{j}) = λ};

(c) the eigenspace of A_{f}^{T} for λ ∈ K is

Span_{K}{(γ(q_{j}, b_{1}, b), ⋯, γ(q_{j}, b_{s}, b)^{T}) : f(α_{j}) = λ};

(d) [q_{j}]f(α_{j}) = [fq_{j}] for each j;

(e) for each λ ∈ K, J : (f − λ) = J iff λ ∉ {f(α_{i}) : 1 ≤ i ≤ s};

(f) for each λ ∈ K J : (f − λ) = J + {q_{j} : f(α_{j}) = λ}.

If, moreover, Y = ∑_{h} c_{h}Z_{h} satisfies condition (AS.1) then:

(4) the j^{th} column (b_{1}(α_{j}), …, b_{s}(α_{j}))^{T} of M_{bq} is the eigenvector for β_{j} := ∑_{h} c_{h}a_{h}^{(j)} of A_{Y};

(5) the j^{th} row (γ(q_{i}, b_{1}, b), ⋯, γ(q_{j}, b_{s}, b))^{T} of M_{qb} is the eigenvector for β_{j} := ∑_{h} c_{h}a_{h}^{(j)} of A_{Y}^{T};

(6) J : (Y − β_{j}) = J + {q_{j}} for each j.

If further b = {[1], [Z_{1}], … [Z_{δ−1}], [b_{δ+1}], ⋯, b_{s}]} satisfies condition (AS.2) then:

(7) denoting {(d_{j1}, …, d_{js})^{T} , 1 ≤ j ≤ s} the eigenvectors of A_{Y} and

α_{j} := (d_{j1}^{−1}d_{j2}, …, d_{j1}^{−1}d_{jδ}, c_{δ0} + ∑_{l=1}^{δ−1} c_{δl}d_{j1}^{−1}d_{jl}, …, c_{n0} + ∑_{l=1}^{δ−1} c_{nl}d_{j1}^{−1}d_{jl})

for each j, then Z(J) = {α_{j}, 1 ≤ j ≤ s}.

Proof. (1), (3)(a-b) are a direct consequence of Lemma 40.2.3, (2), (3)(c-f) of Lemma 40.2.4.

For the non-derogatory case, (4) is Theorem 40.2.6 and (5-6) Corollary 40.2.7.

(7) is a direct reformulation of (1) applied to the basis satisfying condition

(AS.2).  �

Example 40.3.2. If we consider the ideal J ⊂ ℂ[Z_{1}, Z_{2}, Z_{3}] discussed in Example 39.2.3, since we completely know both the roots and the Gröbner structure all we need to do is to verify Auzinger–Stetter’s Theorem on it.

The natural choices for B and 𝕃 are (compare Examples 33.2.5 and 33.2.6)

B := {1, Z_{1}, Z_{2}, Z_{3}, Z_{1}^{2}, Z_{1}Z_{2}, Z_{2}^{2}, Z_{1}Z_{3}, Z_{3}^{2}}

and λ_{i}(p) := p(b_{i}) for all i; under this choice

M_{bq} = ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 2 & 0 & 1 & 1 & 1 & 2 & 2 \\ 0 & 1 & 0 & 2 & 0 & 1 & 1 & 0 & 0 \\ 1 & −2 & 2 & −2 & 3 & 3 & 1 & 1 & 0 \\ 0 & 0 & 4 & 0 & 1 & 1 & 1 & 4 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 4 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 4 & 0 & 3 & 3 & 1 & 2 & 0 \\ 1 & 4 & 4 & 4 & 9 & 9 & 1 & 1 & 0 \end{matrix} );

and the matrices related to the Gröbner representation are (compare Examples 33.5.1 and 33.5.2)

A_{1} = ( \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & −2 & 0 & 0 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 2 & −3 & −9 & −2 & 1 & 6 & 3 & 3 & 0 \\ 6 & −3 & −45 & −8 & 0 & 30 & 15 & 4 & 2 \end{matrix} )

A_{2} = ( \begin{matrix}0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ −2 & −3 & 7 & 2 & 2 & −1 & −3 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & −2 & 0 & 0 & 0 & 3 & 0 & 0 \\ −2 & −3 & 9 & 2 & 2 & −3 & −3 & −1 & 0 \\ −8 & −12 & 40 & 8 & 8 & −19 & −12 & −4 & 0 \end{matrix} )

A_{3} = ( \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ −2 & −3 & 7 & 2 & 2 & −1 & −3 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 2 & −3 & −9 & −2 & 1 & 6 & 3 & 3 & 0 \\ −2 & −3 & 9 & 2 & 2 & −3 & −3 & −1 & 0 \\ −2 & −3 & 9 & 2 & 2 & −1 & −5 & −1 & 0 \\ 6 & −3 & −45 & −8 & 0 & 30 & 15 & 4 & 2 \\ −6 & 3 & −9 & 4 & 0 & 6 & 3 & −3 & 3 \end{matrix} ) .

They satisfy

A_{1}M_{bq} = M_{bq} diag(0, 0, 2, 0, 1, 1, 1, 2, 2)
A_{2}M_{bq} = M_{bq} diag(0, 1, 0, 2, 0, 1, 1, 0, 0)
A_{3}M_{bq} = M_{bq} diag(1, −2, 2, −2, 3, 3, 1, 1, 0)

and none is non-derogatory. Instead the matrix

A_{Y} = ( \begin{matrix}0 & −3 & 1 & 3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −3 & 1 & 0 & 3 & 0 \\ −6 & −9 & 21 & 6 & 6 & −6 & −8 & −3 & 0 \\ −2 & −3 & 7 & 2 & 2 & −1 & −3 & −4 & 3 \\ 6 & −3 & −27 & −6 & −6 & 19 & 9 & 9 & 0 \\ −6 & −9 & 27 & 6 & 6 & −11 & −9 & −3 & 0 \\ −6 & −9 & 25 & 6 & 6 & −6 & −12 & −3 & 0 \\ 10 & −3 & −99 & −16 & −1 & 69 & 33 & 2 & 6 \\ −44 & 6 & 148 & 44 & 8 & −91 & −48 & −25 & 3 \end{matrix} )

related to the linear form Y = −3Z_{1}+Z_{2}+3Z_{3} is non-derogatory and satisfies

A_{Y} M_{bq} = M_{bq} diag(3, −5, 0, −4, 6, 7, 1, −3, −6).

�

40.4 *Möller: Endomorphisms and dual space

If J is radical, setting b := q and recalling that q is the Lagrange basis for the functionals λ_{j} representing the evaluation at α_{j},

λ_{i}(q_{j}) = δ_{ij} = \{ \begin{matrix}1 & if i = j \\ 0 & if i ≠ j. \end{matrix}

we can reformulate Lemma 40.2.3 as

Corollary 40.4.1. For each λ_{j} ∈ 𝕃 the vector

(q_{1}(α_{j}), …, q_{s}(α_{j}))^{T} = (δ_{1j}, …, δ_{sj})^{T} = (0, …, 0, 1, 0, …, 0)^{T}

is an eigenvector of the matrix Q_{f} for the eigenvalue f(α_{j}).

Proof. For each j, 1 ≤ i ≤ s, we have

[fq_{j}] = Φ_{f}([q_{j}]) = ∑_{i} λ_{i}(fq_{j})[q_{i}].  (40.1)

In the particular case of a radical ideal, where each λ_{i} is an evaluation at the point α_{i} we further have

λ_{i}(fq_{j}) = λ_{i}(f)λ_{i}(q_{j}) = f(α_{i})q_{j}(α_{i}) = \{ \begin{matrix}f(α_{j}) & if i = j \\ 0 & if i ≠ j \end{matrix}

so that [fq_{j}] = Φ_{f}([q_{j}]) = ∑_{i} λ_{i}(fq_{j})[q_{j}] = f(α_{j})[q_{j}].  �

In order to extend the argument above to the general case, one needs to expand Equation (40.1) to a generic dual basis 𝕃 and to its corresponding Lagrange basis q.

The natural choice is to take as 𝕃 a Macaulay representation^{2}. Following the notation of Chapter 31, for each τ = Z_{1}^{e_{1}} ⋯ Z_{r}^{e_{r}} ∈ W, we consider the functionals

M(τ) : Q → K, f = ∑_{t∈W} c(f, t)t ↦ c(f, τ)

and we restrict ourselves to the set of Noetherian equations Span_{K}(𝕄) ⊂ Q^{∗} where 𝕄 = {M(τ) : τ ∈ W}.

2 Compare Section 32.1, Corollary 32.3.3, and Definition 33.2.2.

For any term-ordering <, for each ℓ = ∑_{t∈W} c(ℓ, t)M(t) we set T_{<}(ℓ) := min_{<}{t : c(ℓ, t) ≠ 0}, L_{<}(ℓ) := max_{<}{t : c(ℓ, t) ≠ 0} and we denote T_{<}(L) = {T_{<}(ℓ) : ℓ ∈ L) and L_{<}(L) = {L_{<}(ℓ) : ℓ ∈ L) for each subset L ⊂ Span_{K}(𝕄).

We define, for each τ ∈ W, the linear map

σ_{τ} : Span_{K}(𝕄) → Span_{K}(𝕄) : M(ω) ↦ \{ \begin{matrix}M(υ) & if ω = τυ \\ 0 & if τ ∤ ω; \end{matrix}

and, by linearity, for each f = ∑_{t∈W} c(f, t)t ∈ Q the linear map

σ_{f} : Span_{K}(𝕄) → Span_{K}(𝕄), ℓ ↦ σ_{f}(ℓ) = ∑_{t∈W} c(f, t)σ_{t}(ℓ).

We recall that a subspace L ⊂ 𝕄 is called stable (Definition 31.2.2) iff for each ℓ ∈ L, f ∈ Q, σ_{f}(ℓ) ∈ L.

Recall that, for each primary ideal 𝔮 ⊂ P at the origin, the corresponding dual space 𝔏(𝔮) ⊂ Q^{∗} is a stable subset of Span_{K}(𝕄) (Corollary 31.3.3, Proposition 31.3.5) and satisfies T_{<}(𝔏(𝔮)) = N_{<}(𝔮) (Corollary 32.1.4); both T_{<}(𝔏(𝔮)) and L_{<}(𝔏(𝔮)) are ordered ideals.

Moreover it holds Leibnitz Formula

Lemma 40.4.2. For any f, g ∈ Q and any ℓ ∈ Span_{K}(𝕄) we have

ℓ(fg) = ∑_{τ∈W} M(τ)(f)σ_{τ} (ℓ)(g).

Proof. Compare Corollary 31.4.2.  �

An alternative description of the dual space of a primary is in terms of differential functions (Section 31.5): we denote, for each τ = Z_{1}^{e_{1}} ⋯ Z_{r}^{e_{r}} ∈ W, by D(τ) : Q → Q the differential operator

D(τ) := \frac{1}{e_{1}! … e_{r}!} \frac{∂^{e_{1}+···+e_{r}}}{∂Z_{1}^{e_{1}} … ∂Z_{r}^{e_{r}}}

and we consider the subset Span_{K}(𝔻) ⊂ Hom(Q, Q) , 𝔻 = {D(τ) : τ ∈ W}. There is an obvious identification

ev : Span_{K}(𝔻) → Span_{K}(𝕄) : 𝔻(τ) ↦ 𝕄(τ),

which satisfies, for each δ := ∑_{t∈W} c(δ, t)D(t) ∈ Span_{K}(𝔻)

ev(δ)(·) = δ(·)(0, …, 0) = ∑_{t∈W} c(δ, t)M(t)(·)  (40.2)

under which we can impose on 𝔻 the same term-ordering < as induced on 𝕄 so that D(τ) ≤ D(ω) ⟺ M(τ) ≤ M(ω) ⟺ τ ≤ ω and we can set T_{<}(δ) := T_{<}(ev(δ)), L_{<}(δ) := L_{<}(ev(δ)) for each δ ∈ Span_{K}(𝔻) and T_{<}(D) = T_{<}(ev(D))), L_{<}(D) = L_{<}(ev(D))) for each subset D ⊂ Span_{K}(𝔻).

Under this identification we can naturally define the anti-differential operators

σ_{τ} (D(ω)) := \{ \begin{matrix}D(υ) & if ω = τυ \\ 0 & if τ ∤ ω \end{matrix} for each τ, ω ∈ W

and σ_{f}(δ) = ∑_{t∈W} c(f, t)σ_{t}(δ) for each f = ∑_{t∈W} c(f, t)t ∈ Q, δ ∈ Span_{K}(𝔻); a subspace D ⊂ 𝔻 is called stable (Definition 31.5.3) iff σ_{f}(δ) ∈ D for each δ ∈ D, f ∈ Q.

For each α = (a_{1}, … a_{r}) ∈ K^{r} denoting 𝔪_{α} = (Z_{1} − a_{1}, …, Z_{r} − a_{r}) the maximal ideal at α and

λ_{α} : Q → Q, λ_{α}(f) = f(Z_{1} + a_{1}, …, Z_{r} + a_{r})

so that λ_{α}(𝔪_{α}) = m = (Z_{1}, …, Z_{r}) is the maximal at the origin, then we have

(δ(f))(α) = (δ(f))(a_{1}, … a_{r}) = δ(λ_{α}(f))(0, …, 0) = ev(δ)(λ_{α}(f))

for each δ ∈ Span_{K}(𝔻) and f ∈ Q; thus for each 𝔪_{α}-primary 𝔮_{α} if we denote

𝔇_{𝔪_{α}}(𝔮_{α}) := {δ ∈ Span_{K}(𝔻) : δ(f)(α) = 0 for each f ∈ 𝔮_{α}} ⊂ Span_{K}(𝔻)

then we have ev(𝔇_{𝔪_{α}}(𝔮_{α})) = 𝔏(λ_{α}(𝔮_{α})) and 𝔇_{𝔪_{α}}(𝔮_{α}) is stable (under antidifferentiation).

Under this notation Leibnitz Formula becomes

Corollary 40.4.3. For any f, g ∈ Q and any δ ∈ Span_{K}(𝔻) we have

δ(fg) = ∑_{τ∈W} D(τ)(f)σ_{τ} (δ)(g).

We have now the tools needed to describe the dual basis 𝕃 := {λ_{1}, …, λ_{s}} of J and the matrices Q_{f} describing the effect of each endomorphism Φ_{f} in terms of its corresponding Lagrange basis q: using a notation similar to the one used in Section 33.2 we set

• < any term-ordering,

• Z(J) := {α_{1}, …, α_{s}} ⊂ K^{r}, α_{i} = (a_{1}^{(i)}, …, a_{r}^{(i)}), s ≤ s,

• for each i, 1 ≤ i ≤ s – 𝔮_{i} the 𝔪_{α_{i}}-primary component of J, so that J = ∩_{i=1}^{s}𝔮_{i};

– s_{i} := deg(𝔮_{i}) so that ∑_{i=1}^{s} s_{i} = s,

– L_{i} := 𝔏(λ_{α_{i}}(𝔮_{i})) ⊂ Span_{K}(𝕄),

– for each υ ∈ N_{<}(λ_{α_{i}}(𝔮_{i}))

ℓ_{υα_{i}} = M(υ) + ∑_{τ∈W} c(τ, ℓ_{υα_{i}})M(τ) ∈ L_{i}

the unique element (Definition 32.1.3) for which ∘ τ ∈ N_{<}(λ_{α_{i}}(𝔮_{i})) = T_{<}(L_{i}) ⟹ c(τ, ℓ_{υα_{i}}) = 0 and ∘ T_{<}(ℓ_{υα_{i}}) = υ so that ∘ (in particular) ℓ_{1α_{i}} = M(1) and ∘ {ℓ_{υα_{i}} : υ ∈ N_{<}(λ_{α_{i}}(𝔮_{i}))} is the Macaulay basis of

L_{i} = Span_{K}{ℓ_{υα_{i}} : υ ∈ N_{<}(λ_{α_{i}}(𝔮_{i}))};

• 𝕃 := {λ_{1}, …, λ_{s}} := {ℓ_{υα_{i}}λ_{α_{i}} : υ ∈ N_{<}(λ_{α_{i}}(𝔮_{i})), 1 ≤ i ≤ s} ordered so that for λ_{x} := ℓ_{υ_{x}α_{ix}} λ_{α_{ix}} , λ_{y} := ℓ_{υ_{y}α_{iy}} λ_{α_{iy}} holds

x < y ⟺ \{ \begin{matrix}i_{x} < i_{y} & or \\ i_{x} = i_{y} & and υ_{x} < υ_{y}; \end{matrix}

• N_{i} := {h : λ_{h} = ℓ_{τα_{i}}λ_{α_{i}}} = {n_{i}, …, n_{i+1} − 1} for each i, 1 ≤ i ≤ s, where n_{1} := 1, n_{i+1} := 1 + ∑_{l=1}^{i} s_{l}, n_{r+1} = 1 + s;

• q = {q_{1}, …, q_{s}} the set biorthoginal to 𝕃 so that λ_{i}(q_{j}) = δ_{ij};

• for each h, δ_{h} ∈ 𝔻 the element such that λ_{h} = ev(δ_{h})λ_{α_{i}}, h ∈ N_{i}.

We recall that, under these assumptions

(1) [p] = ∑_{i} λ_{i}(p)[q_{i}] for each p ∈ Q;

(2) J_{σ} = {f ∈ Q : λ_{i}(f) = 0, 1 ≤ i ≤ σ} is an ideal for each σ ≤ s, and

(3) J_{1} ⊃ J_{2} ⊃ ⋯ ⊃ J_{s} (cf. Corollary 32.3.3);

(4) by definition, for each i and each f ∈ Q,

ℓ_{1α_{i}}λ_{α_{i}}(f) = M(1)λ_{α_{i}}(f) = λ_{α_{i}}(f)(0, …, 0) = f(α_{i}).

Lemma 40.4.4 (Möller). For each λ_{j} = ℓ_{υα_{i}}λ_{α_{i}} ∈ 𝕃, denoting

T_{j} := {τ ∈ N_{<}(λ_{α_{i}}(𝔮_{i})) : τ < υ} \ {1}

the following holds:

(1) for each f, g ∈ Q, it holds

λ_{j}(fg) = δ_{j}(fg)(α_{i}) = f(α_{i})δ_{j}(g)(α_{i}) + ∑_{τ∈T_{j}} D(τ)(f)σ_{τ} (δ_{j})(g)(α_{i})
= f(α_{i})λ_{j}(g) + ∑_{τ∈T_{j}} M(τ)(f)σ_{τ} (ℓ_{υα_{i}})λ_{α_{i}}(g)

= f(α_{i})λ_{j}(g) + ∑_{x=n_{i}}^{j−1} c_{x}λ_{x}(g)

= f(α_{i})δ_{j}(g)(α_{i}) + ∑_{x=n_{i}}^{j−1} c_{x}δ_{x}(g)(α_{i})

for suitable c_{x} ∈ K; (2) If f = Z_{ι}, for each g ∈ Q

λ_{j}(Z_{ι}g) = δ_{j}(Z_{ι}g)(α_{i}) = a_{ι}^{(i)}δ_{j}(g)(α_{i}) + σ_{Z_{ι}}(δ_{j})(g)(α_{i})
= a_{ι}^{(i)}λ_{j}(g) + σ_{Z_{ι}}(ℓ_{υα_{i}})λ_{α_{i}}(g).

(3) If λ ∈ L_{i} is such that λ(Z_{ι}g) = a_{ι}^{(i)}λ(g), for each g ∈ Q and each ι, 1 ≤ ι ≤ r, then λ = λ_{α_{i}}.

Proof.

(1) A direct consequence of Leibnitz Formula (Lemma 40.4.2 and Corollary 40.4.3) and of the remark that

σ_{τ} (ℓ_{υα_{i}}) ∈ Span_{K}{ℓ_{ςα_{i}} : deg(ς) < deg(υ)} for each τ, υ.

(2) D(τ)(Z_{ι}) = M(τ)(Z_{ι}) = \{ \begin{matrix}1 & if τ = Z_{ι} \\ 0 & if τ ≠ Z_{ι}. \end{matrix}

(3) For λ = ℓλ_{α_{i}}, ℓ = ∑_{τ∈W} c(τ, ℓ)M(τ) ∈ Span_{K}(𝕄), the assumption is equivalent to σ_{Z_{ι}}(ℓ) = 0 for each ι and, in turn, to

c(τ, ℓ) ≠ 0 ⟹ Z_{ι} ∤ τ for each ι
id est ℓ = M(1) = ℓ_{1α_{i}}.  �

Corollary 40.4.5 (Möller). For A_{f} = M([f], b), we have

(1) for each l, 1 ≤ l ≤ s, i, 1 ≤ i ≤ s, j ∈ N_{i}, λ_{j} := ℓ_{υα_{i}}λ_{α_{i}} satisfies

A_{f}λ_{j}(b_{l}) = A_{f}δ_{j}(b_{l})(α_{i}) = f(α_{i})δ_{j}(b_{l})(α_{i}) + ∑_{x=n_{i}}^{j−1} c_{x}δ_{x}(b_{l})(α_{i})

= f(α_{i})λ_{j}(b_{l}) + ∑_{x=n_{i}}^{j−1} c_{x}λ_{x}(b_{l})

for suitable c_{x} ∈ K.

(2) In particular, for each i,

A_{f}λ_{n_{i}}(b_{l}) = A_{f}δ_{n_{i}}(b_{l})(α_{i}) = f(α_{i})δ_{n_{i}}(b_{l})(α_{i}) = f(α_{i})λ_{n_{i}}(b_{l}), 1 ≤ l ≤ s.

(3) On the other side for each i, 1 ≤ i ≤ s, and each j ∈ N_{i}, j ≠ n_{i}, there is at least an l, 1 ≤ l ≤ s, and a ι, 1 ≤ ι ≤ r, for which

A_{ι}λ_{j}(b_{l}) ≠ a_{ι}^{(i)}λ_{j}(b_{l}).

�

Proof. (1) follow by applying λ_{j} to each equation ∑_{i} γ(fb_{l}, b_{i}, b)[b_{i}] = A_{f}[b_{l}] = [fb_{l}] and expanding λ_{j}(fb_{l}) via Lemma 40.4.4.(1).

(2) is the special case j = n_{i} in which the summation is empty.

(3) follow directly from Lemma 40.4.4.(3).  �

Denote, for each ρ ∈ ℕ,

∇_{ρ} := Span_{K} (M(τ), deg(τ) ≤ ρ) and Δ_{ρ} := Span_{K} (D(τ), deg(τ) ≤ ρ)

and set ∇ := ∇_{1} \ ∇_{0} = Span_{K} (M(Z_{h}), 1 ≤ h ≤ r) , and

Δ := Span_{K} (D(Z_{h}), 1 ≤ h ≤ r) = Span_{K} (\frac{∂}{∂Z_{h}} , 1 ≤ h ≤ r) .

Proposition 40.4.6 (Möller–Stetter). The following holds

(1) for δ := ∑_{h=1}^{r} a_{h} \frac{∂}{∂Z_{h}} ∈ Δ and each linear form g = ∑_{h=1}^{r} b_{h}Z_{h} ∈ Q it holds δ(g) = ev(δ)(g) = ∑_{h=1}^{r} a_{h}b_{h} and ev(δ) = ∑_{h=1}^{r} a_{h}M(Z_{i}) ∈ ∇;

(2) for each i, 1 ≤ i ≤ s, ℓ ∈ L_{i} ∩ ∇, δ = ev(ℓ), g ∈ Q, it holds

A_{f}ℓλ_{α_{i}}(g) = A_{f}δ(g)(α_{i}) = f(α_{i})ℓλ_{α_{i}}(g) + ℓλ_{α_{i}}(f)g(α_{i})
= f(α_{i})δ(g)(α_{i}) + δ(f)(α_{i})g(α_{i});

(3) for each i, 1 ≤ i ≤ s, if dim(L_{i} ∩ ∇) > 1 there are ℓ ∈ L_{i}, δ = ev(ℓ) ∈ Δ which satisfiy, for each g ∈ Q,

A_{f}ℓλ_{α_{i}}(g) = A_{f}δ(g)(α_{i}) = f(α_{i})ℓλ_{α_{i}}(g) = f(α_{i})δ(g)(α_{i});

(4) for each i, 1 ≤ i ≤ s, if dim(L_{i} ∩ ∇) = 1, then δ_{n_{i}+j} ∈ Δ_{j} for each j, 1 ≤ j < s_{i};

(5) for each i, 1 ≤ i ≤ s, and each j, 1 ≤ j < s_{i} , if dim(L_{i} ∩ ∇) = 1 and deg(τ) = x, then σ_{τ} (δ_{n_{i}+j}) = δ_{n_{i}+j−x}.

Proof.

(1) trivial.

(2) A direct application of Lemma 40.4.4.(2).

(3) Let us consider two linearly independent elements ℓ_{1}, ℓ_{2} ∈ L_{i} and set b_{ι} := ℓ_{ι}λ_{α_{i}}(f), ι ∈ {1, 2} If b_{ι} = 0 then ℓ_{ι} satisfies the required formula. If b_{1} ≠ 0 ≠ b_{2} then ℓ := b_{2}ℓ_{1} − b_{1}ℓ_{2} satisfies

ℓλ_{α_{i}}(f) = b_{2}ℓ_{1}λ_{α_{i}}(f) − b_{1}ℓ_{2}λ_{α_{i}}(f) = 0

hence A_{f}ℓλ_{α_{i}}(g) = f(α_{i})ℓλ_{α_{i}}(g) for each g ∈ Q.

(4) Let ≺ be any degree-compatible term-ordering. Then for each ℓ ∈

Span_{K}(𝕄),

ℓ ∈ Δ_{ρ} \ Δ_{ρ−1} ⟺ deg(L_{≺}(ℓ)) = ρ.

Since L_{≺}(L_{i}) is an ordered ideal if, for some ρ ∈ ℕ, dim(L_{i} ∩ ∇_{ρ}) > 1 then dim(L_{i} ∩ ∇) > 1.

(5) is a direct consequence of (4).  �

Corollary 40.4.7 (Möller–Stetter). For each i, 1 ≤ i ≤ s, j, 1 ≤ j < s_{i} and each f, g ∈ Q, if dim(L_{i} ∩ ∇) = 1, it holds

λ_{n_{i}+j}(fg) = δ_{n_{i}+j}(fg)(α_{i})
= f(α_{i})λ_{n_{i}+j}(g) + ∑_{x=0}^{j−1} λ_{n_{i}+j−x}(f)λ_{n_{i}+x}(g)

= f(α_{i})δ_{n_{i}+j}(g)(α_{i}) + ∑_{x=0}^{j−1} δ_{n_{i}+j−x}(f)(α_{i})δ_{n_{i}+x}(g)(α_{i}).

Proof. It is a reformulation of Lemma 40.4.4.(1) via Proposition 40.4.6(5). �

40.5 Möller–Stetter: the general case

Let us now consider the general case in which J is not radical and some roots are not simple.

With the notation of Sections 40.2 and 40.4 let us now also set

• Z(J) := {α_{1}, …, α_{s}} ⊂ K^{r}, α_{i} = (a_{1}^{(i)}, …, a_{r}^{(i)}), s ≤ s;

• for each i, 1 ≤ i ≤ r, – λ_{α_{i}} : Q → Q the translation λ_{α_{i}}(Z_{j}) = Z_{j} + a_{j}^{(i)}, for each j,

– 𝔪_{α_{i}} = (Z_{1} − a_{1}^{(i)}, …, Z_{r} − a_{r}^{(i)}), the maximal ideal at α_{i},

– 𝔮_{i} the 𝔪_{α_{i}}-primary component of J, so that J = ∩_{i=1}^{s}𝔮_{i};

– s_{i} := mult(α_{i}, J) = deg(𝔮_{i}) = dim_{K}(L_{i}) the multiplicity in J of α_{i} so that s = ∑_{i=1}^{s} s_{i},

– L_{i} := 𝔏(λ_{α_{i}}(𝔮_{i})) ⊂ Span_{K}(𝕄),

– n_{i+1} := 1 + ∑_{l=1}^{i} s_{l} so that n_{1} = 1 and n_{r+1} = s + 1;

– N_{i} := {n_{i}, …, n_{i+1} − 1};

• 𝕃 := {λ_{1}, …, λ_{s}} the basis biorthogonal to q defined in Section 40.4 so that, in particular, for each i, L_{i} = Span_{K}{λ_{j} : j ∈ N_{i}}, λ_{n_{i}} is the evaluation at α_{i} and λ_{i}(q_{j}) = δ_{ij} for each i, j;

• for each i and each h ∈ N_{i}, δ_{h} ∈ 𝔻 the element such that λ_{h}(·) = δ_{h}(·)(α_{i});

• v_{lj}(b) := λ_{j}(b_{l}) = δ_{j}(b_{l})(α_{i}), 1 ≤ l, j ≤ s, j ∈ N_{i};

• v_{j}(b) = (λ_{j}(b_{1}), …, λ_{j}(b_{s}))^{T} = (δ_{j}(b_{1})(α_{i}), …, δ_{j}(b_{l})(α_{i})^{T} , 1 ≤ j ≤ s, j ∈ N_{i};

• U(b) the square s × s matrix U := (v_{lj}(b)).

Proposition 40.5.1 (Möller–Stetter). Under this notation it holds

(1) U(b) = M_{bq};

(2) each matrix A_{f} = M([f], b) satisfies the relation A_{f}U(b) = U(b)Q_{f}; (3) each matrix Q_{f} = M([f], q) = (q_{lj}) is a block diagonal matrix where

the i^{th} diagonal block, 1 ≤ i ≤ s, is an upper-triangular s_{i} × s_{i} square matrix U_{i} whose diagonal entries are f(α_{i}) and which covers the rows and columns indexed by N_{i} := {n_{i}, …, n_{i+1} − 1} for each i, 1 ≤ i ≤ s.

(4) In particulal, under the assumption dim(L_{i} ∩ ∇) = 1, it holds

q_{ικ} = \{ \begin{matrix}δ_{ι−κ}(f)(α_{i}) & if n_{i} ≤ ι ≤ κ < n_{i+1} \\ 0 & otherwise. \end{matrix}

(5) For each i, 1 ≤ i ≤ s,

v_{n_{i}}(q) = (δ_{1n_{i}}, …, δ_{sn_{i}}))^{T} = (0, …, 0, 1, 0, …, 0)^{T}

is an eigenvector for f(α_{i}) of Q_{f} for each f ∈ Q.

(6) For each i, 1 ≤ i ≤ s,

v_{n_{i}}(b) = (b_{1}(α_{i}), …, b_{s}(α_{s}))^{T}

is an eigenvector of A_{f} for f(α_{i}) for each f ∈ Q.

(7) For b = {[1], [Z_{1}], … [Z_{δ−1}], [b_{δ+1}], ⋯, [b_{s}]} satisfying condition (AS.2) and c_{ιl} ∈ K, 0 ≤ l < δ ≤ ι ≤ r are such that [Z_{ι}] = c_{ι0} + ∑_{l=1}^{δ−1} c_{ιl}[Z_{l}], then for each i, 1 ≤ i ≤ s, if (d_{i1}, …, d_{is})^{T} is an eigenvectors for f(α_{i}) of a non-derogatory matrix A_{f}, f ∈ Q, then

α_{i} := (d_{i1}^{−1}d_{i2}, …, d_{i1}^{−1}d_{iδ}, c_{δ0} + ∑_{l=1}^{δ−1} c_{δl}d_{i1}^{−1}d_{il}, …, c_{n0} + ∑_{l=1}^{δ−1} c_{nl}d_{i1}^{−1}d_{il}) .

(8) [q_{n_{i}}]f(α_{i}) = [fq_{n_{i}}] for each i, 1 ≤ i ≤ s;

(9) for each f ∈ Q and λ ∈ K, J : (f − λ) = J iff λ ∉ {f(α_{i}) : 1 ≤ i ≤ s};

(10) for each f ∈ Q if A_{f} is non-derogatory then J : (f − f(α_{i})) = J + {q_{n_{i}}} for each i, 1 ≤ i ≤ s.

Proof. (1) and (2) are trivial; (3) is a direct consequence of Corollary 40.4.5 and (4) of Corollary 40.4.7; (5) and (6) are trivial consequences of (3); (7) is a direct reformulation of (6) applied to the basis satisfying condition (AS.2);

(8-10) is Lemma 40.2.4.  �

Corollary 40.5.2. It holds

• the trace of A_{f} is Tr(A_{f}) := ∑_{i=1}^{s} s_{i}f(α_{i});

• the determinant of A_{f} is det(A_{f}) := ∏_{i=1}^{s} f(α_{i})^{s_{i}};

• the characteristic polynomial of A_{f} is χ_{f}(T) := ∏_{i=1}^{s} (T − f(α_{i}))^{s_{i}}

• the minimal polynomial of A_{f} is m_{f}(T) := ∏_{i=1}^{s} (T − f(α_{i}))^{ρ_{i}} where ρ_{i} denotes the characteristic number of 𝔮_{i}.

Proof. All the claims are triviall except the one related to the minimal polynomial which is a consequence of the facts that 𝔮_{𝔦} ⊇ 𝔪_{α_{i}}^{ρ} ⟺ 𝔏(λ_{α_{i}}(𝔮_{𝔦})) ⊆ Δ_{ρ}, the multiplicity μ of the eigenvalue f(α_{i}) in the minimal polynomial is the minimal value for which U_{i} = ker (Φ_{f} − f(α_{i}))^{μ} = 0 and of Lemma 40.4.4.(1).

� Remark 40.5.3 (Monico). For simplicity we have assumed K = K throughout this chapter but the construction and the stated results hold also if K ⫋ K; in this case, clearly conjugate roots have the same multiplicity both in χ_{f} and in m_{f}.

Moreover, as a direct application of the Chinese Remainder Theorem, if χ_{f} := ∏_{i=1}^{r} p_{if}(T)^{r_{i}} is an irreducible factorization in K[T], then

J = ⋃_{i=1}^{r} (J + (p_{if}(f)^{r_{i}}))

is an irreducible primary decomposition in K[Z_{1}, …, Z_{r}].

Of course if f is an allgemeine coordinate the corresponding primary decomposition algorithm is the one proposed by Alonso–Raimondo and reported

in Section 35.5.1.  �

40.6 The Univariate case

As a short intermezzo before discussing derogatoriness, let us briefly show how Auzinger–Stetter reformulates the classical elementary linear algebra results for a univariate polynomial.

For a polynomial

f = X^{s} + ∑_{i=0}^{s−1} a_{i}X^{i} = ∑_{i=1}^{s}(X − ξ_{i})^{s_{i}}, s = ∑_{i=1}^{s} s_{i},

the linear representation of J = (q) is the assignement of the normal basis N(J) := {1, X, …, X^{s−1}} and of the Frobenoius companion matrix

A_{1} = ( \begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 \\ −a_{0} & −a_{1} & −a_{2} & ⋯ & −a_{s−1} \end{matrix} ) .

whose characteristic matrix of course is f so that eigenvalues of A_{1} coincide (up to multiplicity) with the roots of f.

Recalling that ξ_{i} is a root of f of multiplicity s_{i} iff f^{(j)}(ξ_{i}) = 0 for 0 ≤ j < s_{i}, the natural choice for the dual space 𝕃 = {λ_{1}, …, λ_{s}} is

λ_{n_{i}}(p) := p(ξ_{i}), λ_{n_{i}+j}(p) := \frac{p^{(j)}(ξ_{i})}{j!}, 1 ≤ j < s_{i}, 1 ≤ i ≤ s,

where we have set n_{i+1} := 1 + ∑_{l=1}^{i} s_{l}, 0 ≤ i < s and the bihortogonal dual basis q is the associated Lagrange basis.

If f is squarefree then M_{qb} is the Vandermonde matrix

M_{qb} = ( \begin{matrix}1 & 1 & ⋯ & 1 \\ ξ_{1} & ξ_{2} & ⋯ & ξ_{s} \\ ⋮ & ⋮ & ... & ⋮ \\ ξ_{1}^{s−1} & ξ_{2}^{s−1} & ⋯ & ξ_{s}^{s−1} \end{matrix} );

in general, each block of the so called generalized Vandermonde matrix M_{qb} has the shape

( \begin{matrix}1 & 0 & 0 & ⋯ & 0 & ⋯ & 0 \\ ξ_{i} & 1 & 0 & ⋯ & 0 & ⋯ & 0 \\ ξ_{i}^{2} & 2ξ_{i} & 1 & ⋯ & 0 & ⋯ & 0 \\ ξ_{i}^{3} & 3ξ_{i}^{2} & 3ξ_{i} & ⋯ & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ... & ⋮ \\ ξ_{i}^{j} & jξ_{i}^{j−1} & \binom{j}{2}ξ_{i}^{j−2} & ⋯ & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ... & ⋮ \\ ξ_{i}^{s−1} & (s − 1)ξ_{i}^{s−2} & \binom{s−1}{2}ξ_{i}^{s−3} & ⋯ & \binom{s−1}{j−1}ξ_{i}^{s−j} & ⋯ & \binom{s−1}{s_{i}−1}ξ_{i}^{s−s_{i}} \end{matrix} ) .

and is related to the i^{th} diagonal block of Q_{1} which is a classical Jordan block

( \begin{matrix}ξ_{i} & 1 & & & 0 \\ & ξ_{i} & 1 \\ & & ... & ... \\ & & & ... & 1 \\ 0 & & & & ξ_{i} \end{matrix} ) .

Example 40.6.1. For

f = X^{8} − X^{7} − X^{6} + 3X^{5} + 9X^{4} − 3X^{3} − 7X^{2} + X + 2
= (X − 2)(X + 1)^{4}(X − 1)^{3}

we have

A_{1} = ( \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ −2 & −1 & 7 & 3 & −9 & −3 & 5 & 1 \end{matrix} )

whose eigenvalues are 2 (simple), -1 (with mulitiplicity 4) and 1 (with mulitiplicity 3) the generalized Vandermonde matrix is

M_{qb} = ( \begin{matrix}1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 2 & −1 & 1 & 0 & 0 & 1 & 1 & 0 \\ 4 & 1 & −2 & 1 & 0 & 1 & 2 & 1 \\ 8 & −1 & 3 & −3 & 1 & 1 & 3 & 3 \\ 16 & 1 & −4 & 6 & −4 & 1 & 4 & 6 \\ 32 & −1 & 5 & −10 & 10 & 1 & 5 & 10 \\ 64 & 1 & −6 & 15 & −20 & 1 & 6 & 15 \\ 128 & −1 & 7 & −21 & 35 & 1 & 7 & 21 \end{matrix} )

related to the Jordan matrix

J_{1} = ( \begin{matrix}2 \\ & −1 & 1 \\ & & −1 & 1 \\ & & & −1 & 1 \\ & & & & −1 \\ & & & & & 1 & 1 \\ & & & & & & 1 & 1 \\ & & & & & & & 1 \end{matrix} )

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Moreover we have

q = (q_{1}, …, q_{s})^{T} = M_{qb}^{−1}(1, X, …, X^{s−1})^{T}

thus allowing to deduce the Lagrange basis by inverting M_{qb}.

Moreover J_{1} = Q_{1} = M([X], q) is the multiplication matrix of K[X]/J w.r.t. Lagrange basis q.

Example 40.6.2. The Example above is to hard to deal by hand, so let us restrict ourselvs to the easier case f = X^{5} − X^{3} = X^{3}(X + 1)(X − 1) where we have

A_{1} = ( \begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{matrix} ) , M_{qb} = ( \begin{matrix}1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & −1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & −1 & 1 \\ 0 & 0 & 0 & 1 & 1 \end{matrix} )

J_{1} = ( \begin{matrix}0 & 1 \\ & 0 & 1 \\ & & 0 \\ & & & −1 \\ & & & & 1 \end{matrix} ) , M_{qb}^{−1} = ( \begin{matrix}1 & 0 & 0 & 0 & −1 \\ 0 & 1 & 0 & −1 & 0 \\ 0 & 0 & 1 & 0 & −1 \\ 0 & 0 & 0 & −\frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{matrix} );

In M_{qb}^{−1} on reads, along the rows, the Lagrange basis

\{1 − X^{4}, X − X^{3}, X^{2} − X^{4}, −\frac{1}{2}(X^{3} − X^{4}), \frac{1}{2}(X^{3} + X^{4})\}.

�

Finally for q ∈ K[X], the i^{th} diagonal block of Q_{q} related to the root ξ_{i} is

( \begin{matrix}q(ξ_{i}) & q^{(1)}(ξ_{j}) & \frac{q^{(2)}(ξ_{i})}{2} & ⋯ & & \frac{q^{(s_{i}−1)}(ξ_{i})}{(s_{i}−1)!} \\ & q(ξ_{i}) & q^{(1)}(ξ_{j}) & \frac{q^{(2)}(ξ_{i})}{2} & & ⋮ \\ & & ... & ... & ... & ⋮ \\ & & & ... & ... & \frac{q^{(2)}(ξ_{i})}{2} \\ & & & & ... & q^{(1)}(ξ_{i}) \\ 0 & & & & & q(ξ_{i}) \end{matrix} ) .

Example 40.6.3. In Example 40.6.2 for q := 1+X^{2} we have A_{q}M_{qb} = M_{qb}Q_{q} with

A_{q} = ( \begin{matrix}1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{matrix} ) , Q_{q} = ( \begin{matrix}1 & 0 & 1 \\ & 1 & 0 \\ & & 1 \\ & & & 2 \\ & & & & 2 \end{matrix} ) .

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40.7 Derogatoriness

The crucial assumption in Theorem 40.5.1 and Proposition 40.5.1 that Φ_{f} is non-derogatory, while it is easily met by a generic linear form in the radical case, is more involved in the general case.

Example 40.7.1. Continuing Example 40.1.1 we remark that neither Z_{1} non Z_{2} while, being allgemeine coordinates, have a non-derogatory matrix.

On the otherside f = Z_{1} − Z_{2} is non-derogatory; in fact with

A_{f} = ( \begin{matrix}0 & 1 & −1 & 0 & 0 \\ 1 & −1 & −1 & 1 & 0 \\ −1 & 1 & 1 & 0 & −1 \\ 1 & 0 & −1 & 0 & 0 \\ −2 & 1 & 0 & 1 & 1 \end{matrix} ) and J := ( \begin{matrix}−1 & 1 \\ & −1 \\ & & 1 & −1 \\ & & & 1 & −1 \\ & & & & 1 \end{matrix} )

we have M_{bq}J = A_{f}M_{bq}.

Why f is a good choice will be explained in Corollary 40.10.6 below. �

In general however, the commuting family {A_{f} : f ∈ Q} does not possess any non-derogatory matrix as it can be seen in the following, trivial, examples:

Example 40.7.2. Let us consider the ideal

J = (Z_{1}, Z_{2})^{2} = (Z_{1}^{2}, Z_{1}Z_{2}, Z_{2}^{2}) ∈ K[Z_{1}, Z_{2}].

For the generic [f] := [a + bZ_{1} + cZ_{2}], the matrix Φ_{f} is represented, via the

basis \{1, Z_{1}, Z_{2}\}, as Φ_{f} = ( \begin{matrix}a & b & c \\ 0 & a & 0 \\ 0 & 0 & a \end{matrix} ); it has the single eigenevalue a with

multiplicity 3 and the eigenspace Span{(1, 0, 0)^{T} , (0, c, −b)^{T} } except in the

trivial case b = c = 0 .  �

Example 40.7.3. In order to dispell the wrong impression that the bad behaviour of the example above could be justified by the reducibility of J, we repeat the same argument for the irreducible primary ideal

J = (Z_{1}^{2}, Z_{2}^{2}) ∈ K[Z_{1}, Z_{2}].

For the generic [f] := [a + bZ_{1} + cZ_{2} + dZ_{1}Z_{2}], the matrix Φ_{f} is repre-

sented, via the basis \{1, Z_{1}, Z_{2}, Z_{1}Z_{2}\}, as Φ_{f} = ( \begin{matrix}a & b & c & d \\ 0 & a & 0 & c \\ 0 & 0 & a & b \\ 0 & 0 & 0 & a \end{matrix} ); the single

eigenevalue is a with multiplicity 4 and

if b^{2} + c^{2} ≠ 0 the eigenspace is Span{(1, 0, 0, 0)^{T} , (0, c, −b, 0)^{T} }; if b = 0 = c, d ≠ 0 the eigenspace is

Span{(1, 0, 0, 0)^{T} , (0, 1, 0, 0)^{T} , (0, 0, 1, 0)^{T} }.

�

Remark that in both examples all eigenspaces share the same joint eigenvector — (1, 0, 0)^{T} and (1, 0, 0, 0)^{T} respectively.

Example 40.7.4. The same can be said for Example 40.1.1 where (1, 0, 1, 0, 1)^{T} is an eigenvector of A_{1} for 0 and of A_{2} for 1 while (1, 1, 0, 1, 0)^{T} is an eigenvector of A_{1} for 1 and of A_{2} for 0; more in general it is easy to verify that, for each f ∈ Q, (1, 0, 1, 0, 1)^{T} is an eigenvector of A_{f} for f(0, 1) and (1, 1, 0, 1, 0)^{T}

is an eigenvector of A_{f} for f(1, 0).  �

The fact that the whole families share at least the eigenvectors

{v_{n_{i}}(b), 1 ≤ i ≤ r}

is already granted by Proposition 40.5.1(5). But there is something more: the set {A_{i}, 1 ≤ i ≤ n} and so a fortiori {A_{f}, f ∈ Q} cannot share any further eigenvector as a consequence of Corollary 40.4.5(3).

In other words in all these examples, each non trivial common eigenspace for the whole family has dimension 1.

Definition 40.7.5. A commuting familty of matrices 𝔄 is called non-dero-gatory if each joint eigenspace has dimension at most 1:

dim_{K}{v ∈ K^{s} : Av = λv, Bv = μv} ≤ 1 for each λ, μ ∈ K, A, B ∈ 𝔄.

A zero-dimensional ideal J ⊂ Q is called a non-derogatory ideal if there is an endomorphism Φ_{f} : A → A for which the matrix A_{f} is non-derogatory.

�

Corollary 40.7.6. The family {Φ_{ι} : 1 ≤ ι ≤ n} is non-derogatory and

v_{n_{i}}(b) := (b_{1}(α_{i}), …, b_{s}(α_{i}))^{T} , i = 1..r

are joined eigenvectors of all the matrices M([f], b), with associated eigenvalue f(α_{i}).

Proof. A direct consequence of Corollary 40.4.5(3).  �

As a consequence, once the eigenspaces of a matrix A_{Y} , Y = ∑_{j=1}^{r} c_{j}Z_{j}, is obtained, if some eigenspaces have dimension greater than 1^{3} one performs the same computation for different matrices A_{Y_{i}}, Y_{i} = ∑_{j=1}^{r} c_{ij}Z_{j}, i = 2..n, being linearly independent forms, and repeatedly applies the eigenspace intersection method, based on a direct application of Lemma 40.7.7 below, to repeatedly compute eigenspace intersections until each eigenspace has dimension 1.

Lemma 40.7.7. Let M, N be two s-square matrices; λ an eigenvalue of M with associated eigenspace U; {u_{1}, …, u_{l}} an orthonormal basis of U.

Define, for each i, j, 1 ≤ i, j ≤ l, a_{ij} := u_{i}^{T} Nu_{j} and set A := (a_{ij}) .

If (d_{1}, …, d_{l})^{T} is any eigenvector of A for μ, then u := ∑_{j} d_{j}u_{j} is a simultaneous eigenvector of M for λ and of N for μ.

3 The possible reasons are two:

(1) either Y is not sufficiently generic and does not satisfy condition (AS.1); in the next steps the variables Y_{i} will separate the roots via eigenspace intersection method;

(2) the ideal is not radical and, even if Y satisfies condition (AS.1), the eigenspace to ∑_{j=1}^{r} c_{j}a_{j}^{(i)} for A_{Y} has dimension greater then 1. The eigenspace intersection method covers also this case thanks of Corollary 40.4.5. However, alternatively, the multiplicity of the roots can be decrased by a proper application of Gianni’s Algorithm (Proposition 35.6.1), e.g. enlarging the ideal J to J + (\sqrt{g(Y)}) where g(A_{Y}) is the characteristic polynomial of A_{Y} (see Remark 40.8.1).

Proof. We have

u is an eigenvector to μ for N

⟺ μ ∑_{j} d_{j}u_{j} = μu = Nu = N ∑_{j} d_{j}u_{j} = ∑_{j} d_{j}Nu_{j}
⟺ μd_{i} = u_{i}^{T} μ ∑_{j} d_{j}u_{j} = ∑_{j} d_{j}u_{i}^{T} Nu_{j} = ∑_{j} a_{ij}d_{j} for each i
⟺ (d_{1}, …, d_{l})^{T} is an eigenvector to μ for A.

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40.8 Stetter Algorithm via Grobnerian Technology

We can assume that the zero-dimensional ideal J is given by means of the Gröbner basis^{4} wrt an ordering <, thus obtaining also the linear representation

N_{<}(J) = {τ_{1}, …, τ_{s}}, M([Z_{h}], N_{<}(J))

thus allowing to compute, with good complexity, the corresponding Gröbner description (cf. Definition 29.3.3)

Rep(g, N_{<}(J)) := (γ(g, τ_{1}, N_{<}(J)), …, γ(g, τ_{s}, N_{<}(J))) ∈ K^{s} :
[g] = ∑_{j} γ(g, τ_{j}, N_{<}(J))[τ_{j}]

for each g ∈ Q.

Remark 40.8.1. Since Stetter Algorithm is improved if J is radical and the matrix A_{Y} is given wrt a linear form Y satisfying condition (AS.1), these results can be efficiently — O(n^{2}s^{3}) — granted by giving an FGLM-like linear algebra version of Gianni’s Proposition 35.6.1 obtained merging the algorithms by Alonso–Raimondo (Algorithm 35.7.1) and Traverso (Algorithm 29.3.8): we choose a linear form Y = ∑_{i} a_{i}Z_{i} and

(1) by linear algebra on the Gröbner descriptions of [1], [Y], [Y^{2}], …, [Y^{s}] compute the minimal polynomial f[Y] ∈ K[Y] such that

f(Y) ∈ J^{+} := J + (Y − ∑_{i} a_{i}Z_{i});

4 An alternative Gröbner-free approach with good complexity for affine complete intersection ideals which gives both a Gröbner representation of J and the corresponding Gröbner description of g, will be discussed in Section 41.15.

The discussion of this section does not require that the obtained representation is linear: Algorithm 35.7.1 equally applies to the data of Section 41.15.

(2) if f is not squarefree, set f := \sqrt{f};

(3) if f is squarefree and d := deg(f) < deg(J) then set j = 1 and (a) while j ≤ n, verify, by linear algebra on the Gröbner descriptions of

[Z_{j}], [1], [Y], [Y^{2}], …, [Y^{d−1}],

whether exists a relation Z_{j} − g_{j}(Y) ∈ J^{+}, deg(g_{j}) < d;

(b) if such a relation exists set j := j + 1 and go to (3.a);

(c) if such relation does not exist i. set J := J + (f(∑_{i} a_{i}Z_{i})) + (Z_{l} − g_{l}(∑_{i} a_{i}Z_{i}), 1 ≤ l < j) ; ii. compute, by linear algebra via Traverso’s Algorithm 29.3.8, the Gröbner basis of J, and deduce the corresponding linear representation and Gröbner descriptions and the value deg J =: s;

iii. set Y := Y + cZ_{j} and go to (1)

(4) if f is squarefree and deg(f) = deg(J), then • J is radical,

• Y satisfies condition (AS.1)

• [Z_{j}] = [g_{j}(Y)] for j = 1 …, n.

�

We can therefore assume of having a linear form Y = ∑_{i} a_{i}Z_{i} satisfying condition (AS.1) and a radical^{5} zero-dimensional ideal J, which is given by means of the Gröbner basis wrt <, and via the linear representation

N_{<}(J) = {τ_{1}, …, τ_{s}}, M([Z_{h}], N_{<}(J))

thus allowing to compute the Gröbner description Rep(g, N_{<}(J)) for each g ∈ Q. Thus, by linear algebra on the Gröbner representations of

[1], [Z_{1}], …, [Z_{r}],

one can obtain with complexity O(ns^{2}), both the K-basis {[1], [Z_{1}], … [Z_{δ−1}]} of V and the linear representations [Z_{i}] = c_{i0} + ∑_{l=1}^{δ−1} c_{il}[Z_{l}], i ≥ δ; further linear algebra on Gröbner representation extends this set to a basis

b = {1, Z_{1}, …, Z_{δ−1}, b_{δ+1}, …, b_{s}} = {b_{1}, …, b_{s}}

satisfying condition (AS.2).

If we denote now 𝕃 := {ℓ_{1}, …, ℓ_{s}} the functionals ℓ_{i}(·) := γ(·, b_{i}, b) so that

[g] = ℓ_{1}(g) + ∑_{i=2}^{δ} ℓ_{i}(g)[Z_{i−1}] + ∑_{i=δ+1}^{s} ℓ_{i}(g)[b_{i}], ∀g ∈ Q

5 Notwithstanding these assumptions A_{Y} is not necessarily non-derogatory; as it is shown by Examples 40.7.2 and 40.7.3 and explained by Corollary 40.10.6 this requires that each primary 𝔮_{i} of J has a good shape.

then 𝕃 is biorthogonal to b; therefore it is sufficient to simply adapt the Enhanced Möller Algorithm (Figure 29.4) to obtain, among the other informations provided by that algorithm, also the matrices M([Z_{i}], b), 1 ≤ i ≤ n. This is all one needs to obtain the matrix M([Y], b) = ∑_{i} a_{i}M([Z_{i}], b).

Once M([Y], b) is obtained, eigenvalue and eigenspace computation is dealt by Numerical Analysis and the joint eigenvectors are obtained, if needed, via the eigenspace intersection method (Lemma 40.7.7).

40.9 Stetter Algorithm

The numerical analysis aspects on the efficiency of the solution of the eigenproblem are out of mine competence^{6} , so I limit myself to note that such efficiency is misured by the condition number

κ(M_{bq}) = ∥M_{bq}∥∥M_{bq}^{−1}∥ = ∥M_{bq}∥∥M_{qb}∥

and is therefore influenced by the arbitrary choise of the basis b; in general κ becomes large if the column vectors v_{j}(b) = (λ_{j}(b_{1}), …, λ_{j}(b_{s}))^{T} are nearly linearly dependent (nearly linearly dependency of rows has naturally the same effect).

It is interesting to remark that for the choice b := N_{≺}(J) wrt a suitable termordering ≺, not surprisingly, the choice of a degree-compatible ordering gives a better condition number than lexicographical orderings.

40.10 Derogatoriness and Allgemeine Coordinates

With the same notation as in Section 40.5, we recall that a linear form

Y := ∑_{h=1}^{r} c_{h}Z_{h}

is said an allgemeine coordinate (Definition 34.4.7) for the zero-dimensional ideal J = ∩_{i=1}^{s}𝔮_{i} iff

(a) there are polynomials g_{i} ∈ K[Y], 0 ≤ i ≤ n, g_{0} monic, deg(g_{i}) < deg(g_{0}), such that

G := (g_{0}(Y), Z_{1} − g_{1}(Y), Z_{2} − g_{2}(Y), …, Z_{r} − g_{r}(Y))

is the reduced Gröbner basis of the ideal

J^{+} := J + (Y − ∑_{h} c_{h}Z_{h}) ⊂ K[Y, Z_{1}, …, Z_{r}]

w.r.t. the lex ordering induced by Y < Z_{1} < … < Z_{r}

6 For that, compare Stetter H., Numerical Polynomial Algebra, SIAM (2004).

and that this condition implies, among the others, that (Corollary 34.4.6)

(b) Q/J ≅ K[Y]/g_{0}(Y)

(c) for each i, 1 ≤ i ≤ s, β_{i} := ∑_{h=1}^{r} c_{h}a_{h}^{(i)} is a root of g_{0} with multiplicity s_{i} and

(d) a_{j}^{(i)} = g_{j}(β_{i}) for each i, 1 ≤ i ≤ s, and each j, 1 ≤ j ≤ r,

(e) g_{0}(Y) = ∏_{i=1}^{r}(Y − β_{i})^{s_{i}};

(f) f ∈ J ⟺ Rem(f(g_{1}(Y), …, g_{r}(Y)), g_{0}(Y)) = 0.

Moreover, there is a Zarisky open set U ⊂ K^{n} such that Y := ∑_{h=1}^{r} c_{h}Z_{h} is an allgemeine coordinate for J iff (c_{1}, …, c_{r}) ∈ U.

For each f = ∑_{t∈W} c(f, t)t : f(0, …, 0) = 0 denote

lin(f) := ∑_{h=1}^{r} c(f, Z_{i})Z_{i}

and, for each primary ideal q at the origin, let

lin{q} := {lin(f) : f ∈ q} and Λ_{1}(q) := {ℓ ∈ ∇_{1} : ℓ(g) = 0, g ∈ lin{q}}

where ∇_{1} denotes ∇_{1} := Span_{K} (M(Z_{i}), 1 ≤ h ≤ r).

Proposition 40.10.1. For each primary ideal q ⊂ Q, deg(q) = s, at the origin the following conditions are equivalent

(1) there is an allgemeine coordinate for q;

(2) dim_{K}(lin{q}) = r − 1;

(3) dim_{K}(Λ_{1}(q)) = 1;

(4) dim_{K}(lin{q}) = r − 1 and for each linear form

Y_{1} := ∑_{h=1}^{r} c_{h}Z_{h} ∉ lin{q}

and each basis (Y_{2}, …, Y_{r}) of lin{q} there are polynomials g_{κ} ∈ K[Y_{1}], 2 ≤ κ ≤ r, such that

q = (Y_{1}^{s}, Y_{2} − g_{2}, …, Y_{r} − g_{r}) ⊂ K[Y_{1}, …, Y_{r}] = K[Z_{1}, …, Z_{r}];

(5) dim_{K}(Λ_{1}(q)) = 1 and for each linear form Y_{1} := ∑_{h=1}^{r} c_{h}Z_{h} ∉ lin{q} and each basis (Y_{2}, …, Y_{r}) of lin{q} there are polynomials g_{κ} ∈ K[Y_{1}], 2 ≤ κ ≤ r, such that

𝔏(q) = \{δ_{l}, 1 ≤ l ≤ s\}, δ_{l}(f) := \frac{f̄^{(l−1)}(0)}{(l − 1)!}, 1 ≤ l ≤ s

where ¯·: Q → Span_{K}{1, Y_{1}, …, Y_{1}^{s−1}} denotes the projection

f(Y_{1}, Y_{2}, …, Y_{r}) ↦ f̄:= Rem(f(Y_{1}, g_{2}(Y_{1}), …, g_{r}(Y_{1})), Y_{1}^{s}); (6) dim_{K}(lin{q}) = r − 1 and for each linear form Y := ∑_{h=1}^{r} c_{h}Z_{h} ∉ lin{q}

there are polynomials g_{κ} = ∑_{l=1}^{s−1} c_{κl}Y^{l} ∈ K[Y], 1 ≤ κ ≤ n, such that q = (Y^{s}, Z_{1} − g_{1}, …, Z_{r} − g_{r});

(7) dim_{K}(Λ_{1}(q)) = 1 and for each linear form Y := ∑_{h=1}^{r} c_{h}Z_{h} ∉ lin{q} there are polynomials g_{κ} ∈ K[Y], 1 ≤ κ ≤ r, such that denoting

˘·: Q ↦ K[Y]/(Y^{s}) : f(Z_{1}, …, Z_{r}) → f̆:= Rem(g_{1}(Y), …, g_{r}(Y)), Y^{s})

we have 𝔏(q) = \{δ_{l}, 1 ≤ l ≤ s\}, δ_{l+1}(f) := \frac{f̆^{(l)}(0)}{l!}, 0 ≤ l < s.

Proof. The scheme of the proof is

(1)

↙  ↖
(2) ↔ (4) → (6)
↕  ↕  ↕
(3)  (5)  (7)

The implications (2) ⟺ (3), (4) ⟺ (5), (6) ⟺ (7), (4) ⟹ (2) and (6) ⟹ (1) hold trivially.

If Y is an allgemeine coordinate for q and g_{κ} = ∑_{l=1}^{s−1} a_{lh}Y^{l}, 1 ≤ h ≤ r are such that

G := (g_{0}(Y), Z_{1} − g_{1}(Y), Z_{2} − g_{2}(Y), …, Z_{r} − g_{r}(Y))

is the reduced Gröbner basis of the ideal

q^{+} := q + (Y − ∑_{h=1}^{r} c_{h}Z_{h})) ⊂ K[Y, Z_{1}, …, Z_{r}]

w.r.t. the lex ordering induced by Y < Z_{1} < … < Z_{r} then

Y ∉ lin{q^{+}} = Span_{K} ({Y − ∑_{h=1}^{r} c_{h}Z_{h}} ∪ {Z_{κ} − a_{1κh}Y), 1 ≤ κ ≤ n})

so that dim_{K}(lin{q}) = dim_{K}(lin{q^{+}}) = 1 and Y = ∑_{h=1}^{r} c_{h}Z_{h} ∉ lin{q}. Thus we have (1) ⟹ (2).

Moreover for Y_{1}, Y_{2}, …, Y_{r} as in (4), the Gröbner basis wrt the lex ordering induced by Y_{1} < Y_{2} < … < Y_{r} of q′ := qK[Y_{1}, …, Y_{r}] necessarily satisfies deg(q′) = deg(q) = s and T(q′) = (Y_{1}^{s}, Y_{2}, …, Y_{r}} thus giving also (2) ⟹ (4).

We are therefore left to prove (4) ⟹ (6).

Let us consider lin{q} ⊂ K[Z_{1}, …, Z_{r}]. There are two cases: either

(i) lin{q} = Span_{K}{Z_{2} − d_{2}Z_{1}, …, Z_{r} − d_{r}Z_{1}} or (ii) there is an Z_{j}, wlog say Z_{1}, such that lin{q} = Span_{K}{Z_{2}, …, Z_{r}}.

In case (i) we set Y_{κ} := Z_{κ} − d_{κ}Z_{1}, 2 ≤ κ ≤ r. By (4) we know that there are

polynomials g_{κ} = ∑_{l=1}^{s−1} c_{κl}Y^{l} ∈ K[Y], 2 ≤ κ ≤ n such that

Y_{κ} − g_{κ}(Y) = Z_{κ} − d_{κ}Z_{1} − c_{κ1}Y + ∑_{l=2}^{s−1} c_{κl}Y^{l} ∈ q

and Z_{κ} − d_{κ}Z_{1} − c_{κ1}Y ∈ lin{q}. Since

(c_{1} + ∑_{κ=2}^{r} c_{κ}d_{κ}) Z_{1} + ∑_{κ=2}^{r} c_{κ}Y_{κ} = ∑_{h=1}^{r} c_{h}Z_{h} = Y ∉ lin{q},

then c := c_{1} + ∑_{κ=2}^{r} c_{κ}d_{κ} ≠ 0 and, setting g_{1} := c^{−1}Y − ∑_{κ=2}^{r} c^{−1}c_{κ}g_{κ} we have

c(Z_{1} − g_{1}) ≡ cZ_{1} − Y + ∑_{κ=2}^{r} c_{κ}Y_{κ} = 0 (mod q) and Z_{1} − g_{1} ∈ q;

(ii) we set Y_{κ} := Z_{κ}, 2 ≤ κ ≤ r and by (4) we know that there are polynomials g_{κ} = ∑_{l=1}^{s−1} c_{κl}Y^{l} ∈ K[Y], 2 ≤ κ ≤ r such that

Y_{κ} − g_{κ}(Y) = Z_{κ} − c_{κ1}Y + ∑_{l=2}^{s−1} c_{κl}Y^{l} ∈ q and Z_{κ} − c_{κ1}Y ∈ lin{q}.

Since Y = c_{1}Z_{1} + ∑_{κ=2}^{r} c_{κ}Z_{κ} ∉ lin{q}, then c_{1} ≠ 0 and c_{κ1} = 0 for each κ > 1; setting g_{1} := c_{1}^{−1}Y − ∑_{κ=2}^{r} c_{1}^{−1}c_{κ}g_{κ} we have

c_{1}(Z_{1} − g_{1}) ≡ c_{1}Z_{1} − Y + ∑_{κ=2}^{r} c_{κ}Y_{κ} = 0 (mod q) and Z_{1} − g_{1} ∈ q.

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Example 40.10.2. Let q := (Z_{2}^{5}, Z_{1} − Z_{2}^{3}) ⊂ K[Z_{1}, Z_{2}] which is a Gröbner basis for the lex ordering indiced by Z_{2} < Z_{1}. Thus

deg(q) = 5, Λ_{1}\{q\} = Span_{K} \{\frac{∂}{∂Z_{2}} \} and lin\{q\} = Z_{1}.

For Y_{1} := aZ_{1} + Z_{2}, Y_{2} = Z_{1} we have q′ = (Y_{1}^{5}, Y_{2} − Y_{1}^{5}). Instead, Z_{1} is not an allgemeine coordinate since, for the lex ordering

indiced by Z_{1} < Z_{2}, we have q = (Z_{1}^{2}, Z_{1}Z_{2}^{2}, Z_{2}^{3} − Z_{1})  �

Remark 40.10.3. As a direct application of Corollary 40.4.7, we have that the functionals δ_{l+1}(f) := \frac{f̆^{(l)}(0)}{l!}, 0 ≤ l < s satisfy

δ_{l+1}(fg)(α_{i}) = f(α_{i})δ_{l+1}(g)(α_{i}) + ∑_{x=0}^{l−1} δ_{1+l−x}(f)(α_{i})δ_{1+x}(g)(α_{i}).

Thus we have the related matrix

Q_{f} = ( \begin{matrix}f(α_{i}) & δ_{2}(f))(α_{i}) & δ_{3}(f)(α_{i}) & ⋯ & & δ_{l+1}(f)(α_{i}) \\ & f(α_{i}) & δ_{2}(f))(α_{i}) & δ_{3}(f)(α_{i}) & & ⋮ \\ & & ... & ... & ... & ⋮ \\ & & & ... & ... & δ_{3}(f)(α_{i}) \\ & & & & ... & δ_{2}(f))(α_{i}) \\ 0 & & & & & f(α_{i}) \end{matrix} ) .

(40.3) �

Definition 40.10.4. A primary ideal q ⊂ Q at the origin which satisfies the equivalent conditions of Proposition 40.10.1 is called a curvilinear primary

with derivative δ_{2}(f) := f̆′(0).  �

For α = (a_{1}, … a_{r}) ∈ K^{r}, denote

λ_{α} : Q → Q, λ_{α}(f) = f(Z_{1} + a_{1}, …, Z_{r} + a_{r})

and 𝔪_{α} = (Z_{1} − a_{1}, …, Z_{r} − a_{r}) the maximal ideal at α.

Theorem 40.10.5. Let J be a zero-dimensional ideal.

Denote Z(J) := {α_{1}, …, α_{s}} ⊂ K^{r}, and, for each i, α_{i} = (a_{1}^{(i)}, …, a_{r}^{(i)}), 𝔮_{i} the 𝔪_{α_{i}}-primary component of J, s_{i} := mult(α_{i}, I) = deg(𝔮_{i}), so that J = ∩_{i=1}^{s}𝔮_{i} and deg(J) = ∑_{i=1}^{s} s_{i} := s.

A linear form Y := ∑_{h=1}^{r} c_{h}Z_{h} is an allegemeine coordinate for J if and only if, denoting β_{i} := ∑_{h=1}^{r} c_{h}a_{h}^{(i)}, the following conditions hold

(1) each primary component 𝔮_{i} of J either (a) is simple so that 𝔮_{i} = 𝔪_{α_{i}} and s_{i} = 1

(b) or the primary ideal q := λ_{α_{i}}(𝔮_{i}) at the origin is curvilinear and

Y ∉ lin{q_{i}};

(2) β_{i} ≠ β_{j} if i ≠ j.

Proof. Assumption (1) implies that for each primary component 𝔮_{i} of J, the ideal

𝔮_{i}^{+} := 𝔮_{i} + (Y − ∑_{h=1}^{r} c_{h}Z_{h}) ⊂ K[Y, Z_{1}, …, Z_{r}]

has a basis (a) (Y − β_{i}, Z_{1} − a_{1}^{(i)}, … Z_{r} − a_{r}^{(i)}) if 𝔮_{i} is maximal; in this case we set

f_{ih}(Z_{1}) := 0 for each h, 1 ≤ h ≤ r;

(b) ((Y − β_{i})^{s_{i}}, Z_{1} − a_{1}^{(i)} − f_{i1}(Y) … Z_{r} − a_{r}^{(i)} − f_{in}(Y)) if 𝔮_{i} is multiple, with f_{ih}(Y) := g_{h}(Y) where g_{h}(Y) are the polynomials whose existence is implied in Proposition 40.10.1(6).

Since, by assumption (2), the β_{i}s are all different, J ∩ K[Y] is generated by g_{0}(Y) = ∏_{j}(Y − β_{i})^{s_{i}}.

By the Chinese Remainder Theorem there is then for each h ≤ 1 a unique polynomial g_{h}(Y), deg(g_{h}) < s = deg(g_{0}) such that

g_{h} ≡ a_{h}^{(i)} + f_{ih} mod (Y − β_{i})^{s_{i}} for each i.

Then (g_{0}(Y), Z_{1} − g_{1}(Y), …, Z_{r} − g_{r}(Y)) is the required Gröbner basis of J^{+} = J + (Y − ∑_{h} c_{h}Z_{h}) implying that Y is an allgemeine coordinate for J.

Conversely if J^{+} has a basis (g_{0}(Y), Z_{1} − g_{1}, …, Z_{r} − g_{r}), then for each primary component 𝔮_{i}, of J, 𝔮_{i}^{+} has a basis (f(Y), Z_{1} −f_{2}, …, Z_{r} −f_{r}) where f runs among the irreducible-power factors of g_{0} and f_{h} = Rem(g_{h}, f) for each h. Thus each component which is not simple has Y as an allgemeine

coordinate.  �

A finer description of derogatoriness of Φ_{f} is the following

Corollary 40.10.6 (Möller–Stetter). Let J be a zero-dimensional ideal and let f ∈ Q. With the notation of Theorem 40.10.5, Φ_{f} is non-derogatory if and only if the following conditions hold:

(1) f(α_{i}) ≠ f(α_{j}) if i ≠ j,

(2) each primary component 𝔮_{i} of J either (a) is simple so that 𝔮_{i} = 𝔪_{α_{i}}

(b) or the primary ideal λ_{α_{i}}(𝔮_{i}) at the origin is curvilinear with derivative

ℓ_{i}

(3) for each multiple component 𝔮_{i}, ℓ_{i}(f) ≠ 0.

Proof. Assume that Φ_{f} is non-derogatory. Then:

(1) for α_{i} ≠ α_{j} we have v_{n_{i}}(b) ≠ v_{n_{j}}(b); thus necessarily f(α_{i}) ≠ f(α_{j}) otherwise the two vectors are independent eigenvectors for f(α_{i}) = f(α_{j}).

(2) For a multiple component 𝔮_{i}, Proposition 40.4.6(3) implies that if L_{i} ∩ ∇ = Λ_{1}(λ_{α_{i}}(𝔮_{i})) has dimension > 1, then there is an eigenvector for f(α_{i}) linearly independent with v_{n_{i}}(b); thus 𝔮_{i} must be curvilinear.

(3) If instead L_{i} ∩ ∇ = Λ_{1}(λ_{α_{i}}(𝔮_{i})) has dimension 1, Corollary 40.4.7 shows that if ℓ_{i}(f) = 0 then v_{n_{i}+1}(b) is a further eigenvector for f(α_{i}).

Conversely, by (2) the matrix Q_{f} has s blocks; the i^{th} one has Equation (40.3) as shape; thus, since, by (3), ℓ_{i}(f) = δ_{2}(f)(α_{i}) ≠ 0, each block contributes a single eigevector; finally (1) grants that different eigenvectors

correspond to different eigenvalues.  �

Remark 40.10.7 (Möller–Stetter). If δ_{2}(f)(α_{i}) = ⋯ = δ_{j}(f)(α_{i}) = 0 ≠ δ_{j+1}(f)(α_{i}) then the i^{th} block (40.3) contributes j linearly independent eigen-

vectors.  �

Corollary 40.10.8. A zero-dimensional ideal is a non-derogatory ideal if each multiple primary component is curvilear.

Proof. Each linear form Y = ∑_{h} c_{h}Z_{h} is non-derogatory provided it satisfies Y (α_{i}) ≠ Y (α_{j}) for each i ≠ j and^{7} δ_{2}(Y) ≠ 0.

Both conditions define a Zarisky open set.  �

7 Since both Y and δ_{2} are linear, then δ_{2}(Y) is a constant.

41. Macaulay IV

The device that follows […] finally eliminates from algebraic geometry the last traces of Elimination Theory A. Weil Eliminate, eliminate, eliminate, Eliminate the eliminators of elimination theory. S.S. Abhyankar

After having discuss the two ‘standard’ recent algorithms for solving, this chapter is devoted to the old fashoned tools of resultants and resolvants.

After recalling the notion and the main properties of resultants (Section 41.1) we mainly discuss Macaulay’s approach^{1} for computing it: Macaulay defines the resultant as the gcd of all determinants of a matrix, Macaulay’s matrix (Section 41.3) obtained by expanding a proper generating set which can be deduced by a result of Bézout (Section 41.2) and proves that the resultant is obtained by dividing out from any such determinant an extraneous factor (Section 41.4) which he is able to precisely characterize; finally we are able to prove that Macaulay’s definition is really the resultant (Section 41.5).

The knowledge of the resultant of a set of forms allows to compute the roots of the ideal

J := 𝕀(f_{1}, …, f_{r}} ⊂ K[Z_{1}, …, Z_{r}]

by computing in K[U_{1}, …, U_{r}][Z] the resultant (u-resultant) of the polynomials f_{1}, …, f_{r}, f_{u}, f_{u} := Z −∑_{i=1}^{r} U_{i}Z_{i} and factorizing it into linear factors Z − ∑_{i=1}^{r} U_{i}α_{i}; each such linear factor returns a root (α_{1}, …, α_{r}) ∈ Z(J) (Section 41.6).

1 Where I mainly follows the original result Macaulay F. S., Some Formulae in Elimination, Proc. London Math. Soc.

(1) 35 (1903), 3–27

supported by his book Macaulay F. S. , The Algebraic Theory of Modular Systems, Cambridge Univ. Press (1916)

I next discuss^{2} another Nineteenth Century solver, Kronecker’s resolvant (Section 41.7); if given an ideal

I := 𝕀(f_{1}, …, f_{s}} ⊂ k[X_{1}, …, X_{n}]

one computes the resolvant (u-resolvant) in k[Λ_{1}, …, Λ_{n}][X_{1}, …, X_{n−1}][X] of the polynomials

Λ_{n}^{deg(f_{l})}f_{l} (X_{1}, …, X_{n−1}, Λ_{n}^{−1} (X − ∑_{i=1}^{n−1} Λ_{i}X_{i})) , 1 ≤ l ≤ s;

and factorizes it into linear factors

X − Λ_{1}X_{1} − ⋯ − Λ_{ν−1}X_{ν−1} − Λ_{ν}ξ_{ν} − ⋯ − Λ_{n}ξ_{n};

each factor for which each ξ_{i} is independent of the Λs corresponds to an Ω(k)-prime component of dimension ν − 1 (Section 41.8).

Notwithstanding Macaulay strongly criticizes Kronecker’s resolvant for its being doubly exponential in comparison with the simply exponential resultant, Kronecker’s approach provided (in the Nineteenth Century!) a parametrization

\{ q(X_{1}, …, X_{ν−1}, U) = 0, ; X_{ν} = \frac{w_{ν}(X_{1},...,X_{ν−1},U)}{\frac{∂q}{∂U} (X_{1},...,X_{ν−1},U)} ; ⋮ ; X_{n} = \frac{w_{n}(X_{1},...,X_{ν−1},U)}{\frac{∂q}{∂U} (X_{1},...,X_{ν−1},U)}  (41.1)

of a radical equidimensional ideal (Section 41.9) which is at the core of the most efficient to-day solvers: Rouillier’ s Rational Univariate Representation (Section 42.9) and TERA’s Kronecker Package (Chapter 44).

After an intermezzo where I cover the history of the resultants from Bézout to the English algebra school (Section 41.10) and, in particular, I report Cayley’s interpretation of the resultant of two polynomials

U(X), V (X) ∈ k[X], deg(U) = deg(V) = n

as the determinant of the matrix (α_{ρσ}) defined by the relation

\frac{U(X)V (Y) − U(Y)V (X)}{X − Y} = ∑_{ρ=0}^{n−1} ∑_{σ=0}^{n−1} α_{ρσ}X^{n−ρ−1}Y^{n−σ−1}

I briefly discuss a different representation of the resultant due to Dixon which extended Cayley’s interpretation to more than 2 polynomials (Section 41.11).

Dixon’s resultant was recently revised and reproposed by Kapur et al.; at the same time, also Cardinal considered Cayley’s formula and Dixon’s matrix

2 Still following the guideline provided by Macualay’s book

41.1 The resulatant of r forms in r variables 65

(Section 41.12) and proposed an algorithm (Section 41.13) which performed a series of transformations on the Dixon’s matriices defined by the polynomials

f_{1}, …, f_{n}, X_{i}, 1 ≤ i ≤ n, f_{l} ∈ k[X_{1}, …, X_{n}];

he conjectured that, if f_{1}, …, f_{n} is a complete intersection the output of his algorithm is a Gröbner representation of the ideal

J := 𝕀(f_{1}, …, f_{n}) ⊂ k[X_{1}, …, X_{n}].

Recently Mourrain proposed an improved version of Cardinal’s algorithm and gave a complete proof of Cardinal’s conjecture for this abridged version of the algorithm (Section 41.14).

The result, not only returns, with good complexity, a Gröbner representation of the ideal J but, with the same complexity allows to test ideal membership and for a polynomial f ∈ J returns also a representation f = ∑_{i=1}^{n} g_{i}f_{i} (Section 41.15).

41.1 The resulatant of r forms in r variables

Let Q := K[Z_{1}, …, Z_{r}], W := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} : (a_{1}, …, a_{r}) ∈ ℕ^{r}} its monomial K-basis and K the algebraic closure of K.

For each d ∈ ℕ we also set

W_{d} := {t ∈ W : deg(t) = d} and W(d) := {t ∈ W : deg(t) ≤ d}.

Let us also set r integers d_{1}, …, d_{r}. Our aim being considering r ’generic’ forms (homogeneous polynomials) f_{1}, …, f_{r} ∈ Q, deg(f_{i}) = d_{i}, we apply the same notation and approach of the English algebra school (cf. Sections 6.4-7).

Since each homogeneous polynomial f ∈ P, deg(f) = d, can be uniquely expressed as f = ∑_{τ∈W_{d}} c(f, τ)τ, we introduce indeterminate coefficients a_{i,τ} , 1 ≤ i ≤ r, τ ∈ W_{d_{i}}, and we consider

the domain 𝔻 := ℤ[a_{i,τ} , 1 ≤ i ≤ r, τ ∈ W_{d_{i}}],

its quotient field 𝕂 := ℚ(a_{i,τ} , 1 ≤ i ≤ r, τ ∈ W_{d_{i}}), and the ’generic’ forms F_{i} = ∑_{τ∈W_{d_{i}}} a_{i,τ} τ, 1 ≤ i ≤ r;

for any set f := {f_{1}, …, f_{r}} of r concrete homogeneous forms

f_{i} = ∑_{τ∈W_{d_{i}}} c(f_{i}, τ)τ

of degree d_{i} we denote Ξ_{f} : 𝔻[Z_{1}, …, Z_{r}] → K[Z_{1}, …, Z_{r}] the ansatz

Ξ_{f} (a_{i,τ}) = c(f_{i}, τ) for each 1 ≤ i ≤ r, τ ∈ W_{d_{i}},

so that Ξ_{f} (F_{i}) = f_{i} for each i.

Definition 41.1.1. A polynomial

Res := Res(d_{1}, …, d_{r}) := Res(F_{1}, …, F_{r}) ∈ 𝔻

is called the resultant of F_{1}, …, F_{r} if for any set f := {f_{1}, …, f_{r}} of r homogeneous forms f_{i} of degree d_{i}

Ξ_{f} (Res) = 0 ⟺ exists α ∈ ℙ^{r−1}(K) : f_{1}(α) = ⋯ = f_{r}(α) = 0.

�

Fix a variable, say Z_{r}, and define a weight on the variables a_{i,τ} by setting wt(a_{i,τ}) = deg_{r}(τ) id est wt(a_{i,τ}) = a_{r} for each τ = Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} .

Set D := ∏_{i=1}^{r} d_{i}, D_{i} := D/d_{i} for each i and

d := 1 − r + ∑_{i=1}^{r} d_{i} = 1 + ∑_{i=1}^{r}(d_{i} − 1).

Fact 41.1.2. With the present notations the following holds:

(1) For each r − 1 homogeneous polynomials f_{1}, …, f_{r−1} ∈ K[Z_{1}, …, Z_{r}] which generate a homogeneous ideal J := (f_{1}, …, f_{r−1}) having only a finite number of (projective) zeroes, we have #Z(J) = D_{r} = ∏_{i=1}^{r−1} d_{i}.

(2) The resultant Res(F_{1}, …, F_{r}) is (a) homogeneous of degree D_{i} in the varaibles a_{i,τ} for each i and

(b) isobaric^{3} of weight D.

Proof.

(1) It is trivial for r = 2 and is obtained, for r > 2, by considering the polynomials f_{i} as elements in K[Z_{1}][Z_{2}, Z_{3}, …, Z_{r}]; thus (2) implies that Res(f_{1}, …, f_{r−1}) ∈ K[Z_{1}] is a univariate polynomial of degree D_{r}.

(2) For r = 2, it is a homogeneous reformulation of the description of the structure of the Sylvester resultant: for

F_{1}(Z_{1}, Z_{2}) = a_{0} ∏_{i=1}^{d_{1}}(Z_{1} − α_{i}Z_{2}) = a_{0}Z_{1}^{d_{1}} + a_{1}Z_{1}^{d_{1}−1}Z_{2} + ⋯ + a_{d_{1}}Z_{2}^{d_{1}},

F_{2}(Z_{1}, Z_{2}) = b_{0} ∏_{j=1}^{d_{2}}(Z_{1} − β_{j}Z_{2}) = b_{0}Z_{1}^{d_{2}} + b_{1}Z^{d_{2}−1}Z_{2} + ⋯ + b_{d_{2}}Z_{2}^{d_{2}},

if we consider the F_{i}s as elements of K[Z_{2}][Z_{1}], (a) comes from Proposition 6.6.8 and (b) from Proposition 6.7.1 which returns

3 A function D ∈ 𝔻 is called isobaric iff all of its terms are of the same weight.

Res(F_{1}, F_{2})
= a_{0}^{d_{2}}b_{0}^{d_{1}} ∏_{i=1}^{d_{1}} ∏_{j=1}^{d_{2}}((α_{i} − β_{j})Z_{2}) = a_{0}^{d_{2}}b_{0}^{d_{1}}Z_{2}^{d_{1}d_{2}} ∏_{i=1}^{d_{1}} ∏_{j=1}^{d_{2}}(α_{i} − β_{j})

= a_{0}^{d_{2}} ∏_{i=1}^{d_{1}} F_{2}(α_{i}Z_{2}) = a_{0}^{d_{2}}Z_{2}^{d_{1}} ∏_{i=1}^{d_{1}} g(α_{i}) =

= (−1)^{d_{1}d_{2}}b_{0}^{d_{1}} ∏_{j=1}^{d_{2}} f(β_{j}Z_{2}) = (−1)^{d_{1}d_{2}}b_{0}^{d_{1}}Z_{2}^{d_{2}} ∏_{j=1}^{d_{2}} f(β_{j});

thus, Res(F_{1}, F_{2}) = Z_{2}^{d_{1}d_{2}}Res(f, g) is obtained by substituting each instance of a_{i}, b_{j} by Z_{2}^{i}a_{i}, Z_{2}^{j}b_{j} respectively. For r > 2, we obtain^{4} (2) by applying (1) to the polynomials f_{1}, …, f_{r−1}, for each ansatz Ξ(F_{i}) = f_{i}, 1 ≤ i < r: denoting

(1, λ_{2}^{(i)}, …, λ_{r}^{(i)}), 1 ≤ i ≤ D_{r} = ∏_{i=1}^{r−1} d_{i}

their roots, let us then define R := ∏_{i=1}^{D_{r}} F_{r}(1, λ_{2}^{(i)}, …, λ_{r}^{(i)}) ∈ 𝔻 which clearly satisfies R = Ξ(Res(F_{1}, …, F_{r})) and which is the numerator of a symmetric function of the roots and thus can be expressed in terms of the coefficients of the f_{i}s. Thus Res(F_{1}, …, F_{r}) is isobaric of degree d_{r}D_{r} = D and is homogeneous of degree D_{r} in the variables a_{r,τ}. �

41.2 Bézout’s Generating Set

Let us denote J := (F_{1}, …, F_{r}) ⊂ 𝔻[Z_{1}, …, Z_{r}] the homogeneous ideal generated by the generic forms F_{i} and for each δ ∈ ℕ,

J_{δ} := J ∩ W_{δ} = {F ∈ J homogeneous, deg(F) = δ},

and let us remark that^{5}

B := {ωF_{i} : ω ∈ W_{δ−d_{i}}, 1 ≤ i ≤ r}

is a 𝔻-generating set of J_{δ}, so that, for each homogeneous polynomial F ∈ J_{δ}, there are homogeneous polynomials

S_{i} = ∑_{ω∈W_{δ−d_{i}}} c(S_{i}, ω)ω ∈ 𝔻[Z_{1}, …, Z_{r}], deg(S_{i}) = δ − d_{i} for each i

4 I limit myself here to sketch Poisson’s proof of this result which is never used in the argument which leeds to Theorem 41.5.3. For a proof of Fact 41.1.2(2) I refer to Remark 41.5.1.

5 With a slight abuse of notations, W_{z} = ∅ for each z ∈ ℤ, z < 0.

such that F = ∑_{i=1}^{r} S_{i}F_{i}; such representations are, of course, not unique.

Uniqueness can be however forced, restricting the spans of the F_{i}s: let us denote, for each i ≤ r + 1,

W_{δ}^{(i)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{i}} : a_{j} < d_{j} for each j < i}

so that

W_{δ}^{(1)} = W_{δ−d_{1}}; W_{δ}^{(2)} consists of all terms of degree δ − d_{2} which are not divisable by Z_{1}^{d_{1}}; W_{δ}^{(3)} consists of all terms of degree δ − d_{3} which are not divisable by Z_{1}^{d_{1}}

nor by Z_{2}^{d_{2}}; … W_{δ}^{(i)} consists of all terms τ of degree δ − d_{i} such that τ ∉ (Z_{1}^{d_{1}}, …, Z_{i−1}^{d_{i−1}}); … W_{δ}^{(r+1)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ} : a_{j} < d_{j} for each j ≤ r}.

Remark 41.2.1. For each δ ≥ d = 1 + ∑_{i=1}^{r}(d_{i} − 1), we have W_{δ}^{(r+1)} = ∅ since for Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ}^{(r+1)} we have the contradiction

1 + ∑_{i} a_{i} ≤ 1 + ∑_{i=1}^{r}(d_{i} − 1) = d ≤ δ = ∑_{i=1}^{r} a_{i}.

�

Lemma 41.2.2. With the notation above, for any ν ≤ r, W_{δ} has the partition

W_{δ} = {τZ_{1}^{d_{1}}, τ ∈ W_{δ}^{(1)}} ⊔ ⋯ ⊔ {τZ_{ν}^{d_{ν}} , τ ∈ W_{δ}^{(ν)}} ⊔ W_{δ}^{(ν+1)}
= V_{δ}^{(1)} ⊔ ⋯ ⊔ V_{δ}^{(i)} ⊔ ⋯ V_{δ}^{(ν)} ⊔ W_{δ}^{(ν+1)}

where, for each i ≤ r,

V_{δ}^{(i)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{i}} : a_{i} ≥ d_{i}, a_{j} < d_{j} for each j < i}.

�

Theorem 41.2.3 (Bézout). For any ν ≤ r, each homogeneous polynomial F ∈ 𝔻[Z_{1}, …, Z_{r}] of degree δ can be uniquely expressed as

ΔF = ∑_{i=1}^{ν} Q_{i}F_{i} + Q_{ν+1}

where Δ ∈ 𝔻, Δ ≠ 0, and

Q_{i} = ∑_{ω∈W_{δ}^{(i)}} c(Q_{i}, ω)ω ∈ 𝔻[Z_{1}, …, Z_{r}] for each i.

Proof. Let us begin by remarking that, by Lemma 41.2.2 above,

∑_{i=1}^{ν+1} #W_{δ}^{(i)} = ∑_{i=1}^{ν} #V_{δ}^{(i)} + W_{δ}^{(ν+1)} = #W_{δ};

this counting argument can be also deduced by considering the polynomial ∑_{i=1}^{ν} Q_{i}Z_{i}^{d_{i}} +Q_{ν+1} where each term of degree δ comes in once and once only.

Equating, for each τ ∈ W_{δ}, the coefficient of τ in F with the one in ∑_{i=1}^{ν} Q_{i}F_{i} + Q_{ν+1} we obtain #W_{δ} equations in the ∑_{i=1}^{ν+1} #W_{δ}^{(i)} = #W_{δ} unknowns c(Q_{i}, ω), ω ∈ W_{δ}^{(i)}, 1 ≤ i ≤ ν + 1.

The corresponding determinant Δ cannot vanish, otherwise we would obtain a non trival solution of the equation ∑_{i=1}^{ν} Q_{i}F_{i} + Q_{ν+1} = 0; it would then be sufficient to make the ansatz F_{i} := Z_{i}^{d_{i}}, 1 ≤ i ≤ ν, in order to deduce a contraddictory identity ∑_{i=1}^{ν} Q_{i}Z_{i}^{d_{i}} + Q_{ν+1} = 0 where some Q_{i} is not van-

ishing. Hence the theorem is proved.  �

Remark 41.2.4. Denote S_{r} the symmetric group of all the permutations π : {1, …, r} → {1, …, r} over {1, …, r}.

The result of Theorem 41.2.3 being independent of the ordering chosen by the variables, for each permutation π ∈ S_{r}, each homogeneous polynomial F ∈ 𝔻[Z_{1}, …, Z_{r}] of degree δ can be uniquely expressed as ΔF = ∑_{i=1}^{ν} Q_{i}F_{π(i)} + Q_{ν+1} where ν ≤ r, Δ ∈ 𝔻, Δ ≠ 0, and Q_{i} ∈ Span_{𝕂}(W_{πδ}^{(i)}) where

W_{πδ}^{(i)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{π(i)}} : a_{π(j)} < d_{π(j)} for each j < i}.

�

41.3 Macaulay’s Matrix

Let us now represent the 𝔻-generating set

B := {ωF_{i} : ω ∈ W_{d−d_{i}}1 ≤ i ≤ r}

of J_{d} by a matrix, the Macaulay’s matrix whose columns are indexed by the \binom{d+r−1}{r−1} terms τ ∈ W_{d} and each of whose rows is indexed by one of the elements

∑_{τ∈W_{d}} c(ωF_{i}, τ)τ = ωF_{i} ∈ B

and has c(ωF_{i}, τ) := \{ \begin{matrix}a_{i,υ} & if τ = υω \\ 0 & if ω ∤ τ \end{matrix} as its τ-entry.

Fig. 41.1. Macaulay’s Matrix

x^{3} x^{2}y x^{2}z xy^{2} xyz xz^{2} y^{3} y^{2}z yz^{2} z^{3}
xF_{1} X x  x  x  x  x 0 0  0 0
yF_{1}  0 X  0  x  x  0 x x  x 0
zF_{1}  0 0  X  0  x  x 0 x  x x
xF_{2} x x  x X x  x 0 0  0 0
yF_{2}  0 x  0  x  x  0 X x  x 0
zF_{2}  0 0  x  0  x  x 0 X x x
x^{2}F_{3} x x  X  0  0  0 0 0  0 0
xyF_{3} 0 x  0  x X  0 0 0  0 0
xzF_{3} 0 0  x  0  x X 0 0  0 0
y^{2}F_{3} 0 0  0  x  0  0 x X  0 0
yzF_{3} 0 0  0  0  x  0 0 x X 0
z^{2}F_{3} 0 0  0  0  0  x 0 0  x X

Example 41.3.1. Let us set Q = K[x, y, z], d_{1} = d_{2} = 2, d_{3} = 1 and δ = 3. In Figure 41.1 we represent such matrix. Each 0 indicates that the entry is 0; each x indicates that the entry is a variable a_{i,υ}. The elements a_{i,Z_{i}^{d_{i}}} are

represented X.  �

Definition 41.3.2 (Macaulay). The (Macaulay’s) resultant of F_{1}, …, F_{r} is the greatest common divisor of all the determinants of the above matrix.

�

Denoting

R := R(d_{1}, …, d_{r}) := R(F_{1}, …, F_{r}) ∈ 𝔻

the Macaulay’s resultant of F_{1}, …, F_{r}, let us consider the determinant D of the square matrix^{6} obtained by selecting the rows indexed by the polynomials in the basis

B := {ωF_{i} : ω ∈ W_{d}^{(i)}, 1 ≤ i ≤ r}

and let us study its properties:

Example 41.3.3. In Figure 41.1 the two rows in sans serif (indexed by x^{2}F_{3} and y^{2}F_{3}) are the ones to be removed in order to obtain the matrix D. �

Proposition 41.3.4 (Macaulay). Setting, for each i ≤ r

a_{i} := a_{i,Z_{i}^{d_{i}}} , c_{i} := a_{i,Z_{r}^{d}}, μ_{i} := #W_{d}^{(i)}

the following holds

(1) c(D, a_{1}^{μ_{1}} ⋯ a_{r}^{μ_{r}}) = ±1;

(2) Ξ_{f} (c_{i}) = Ξ_{f} (a_{i,Z_{r}^{d}}) = c(f_{i}, Z_{r}^{d}) = 0 for each i ⟹ Ξ_{f} (D) = 0;

6 The matrix is square as a consequence of Lemma 41.2.2 and Remark 41.2.1.

(3) for each τ ∈ W_{d}, Dτ ∈ J;

(4) μ_{r} := #(W_{d}^{(r)}) = ∏_{i=1}^{r−1} d_{i} = D_{r};

(5) D is homogeneous of degree D_{r} in the coefficients of F_{r} while

(6) D is homogeneous in the coefficients of F_{i} with a degree > D_{i} := \frac{D}{d_{i}} for each i < r.

(7) D is isobaric with weight D := ∏_{i=1}^{r} d_{i}.

Proof.

(1) Clearly in each row appears one and only one a_{i}; in order to prove the claim we must show that no two such a_{i} can appear in the same column. This is a trivial consequence of two remarks above, namely that W_{d}^{(r+1)} = ∅ and that, making in the expression ∑_{i=1}^{r} Q_{i}F_{i} + Q_{r+1} = ∑_{i=1}^{r} Q_{i}F_{i} the ansatz F_{i} := a_{i}Z_{i}^{d_{i}}, 1 ≤ i ≤ r we obtain ∑_{i=1}^{r} Q_{i}a_{i}Z_{i}^{d_{i}} where each term of degree d comes in once and once only.

(2) The column indexed by Z_{r}^{d} contains all zeros excepts in the rows indexed by Z_{r}^{d−d_{i}}F_{i} where the value is a_{i,Z_{r}^{d_{i}}}.

(3) Denoting D_{i,ω} the subdeterminant obtained crossing the column indexed by τ and the row corresponding to the polynomial ωF_{i} we have

∑_{τ∈W_{d}} Dτ = ∑_{τ∈W_{d}} ∑_{i,ω} D_{i,ω}c(ωF_{i}, τ)τ
= ∑_{i,ω} D_{i,ω} ∑_{τ∈W_{d}} c(ωF_{i}, τ)τ
= ∑_{i,ω} D_{i,ω}ωF_{i}
≡ 0 mod J.

(4) deg(Q_{r}) = d − d_{r} = 1 + ∑_{i=1}^{r}(d_{i} − 1) − d_{r} = ∑_{i=1}^{r−1}(d_{i} − 1) and the terms of Q_{r} consist of the set of all terms in the expansion

∏_{i=1}^{r−1} (Z_{r}^{d_{i}−1} + Z_{i}Z_{r}^{d_{i}−2} + ⋯ + Z_{i}^{d_{i}−2}Z_{r} + Z_{i}^{d_{i}−1})

whence μ_{r} = #(W_{d}^{(r)}) = # supp(Q_{r}) = ∏_{i=1}^{r−1} d_{i} = D_{r}.

(5) Is a trivial consequence of the resut above.

(6) In general, the terms τ = ωυ of Q_{j} consist of the set of all terms ω in the expansion ∏_{i=1}^{j−1} (1 + Z_{i} + ⋯ + Z_{i}^{d_{i}−1}) each multiplied by a term

υ = Z_{j}^{a_{j}} ⋯ Z_{r}^{a_{r}}, deg(υ) = d − d_{j} − deg(ω) so that

μ_{i} = #(W_{d}^{(i)}) = # supp(Q_{i}) ≥ D_{i} = ∏_{{i=1 \atop i≠j}}^{r} d_{i}.

(7) For any element

ωF_{i} = ∑_{υ∈W_{d−d_{i}}^{(r)}} a_{i,υ}ωυ = ∑_{τ∈W_{d}} c(ωF_{i}, τ)τ ∈ B

we have, for each c(ωF_{i}, τ) ≠ 0,

wt(c(ωF_{i}, τ) = wt(a_{i,υ}) = deg_{r} υ = deg_{r} τ − deg_{r} ω;

hence, on expanding D the weight of each term is

∑_{τ∈W_{d}} deg_{r} τ − ∑_{i=1}^{r} ∑_{ω∈W_{d}^{(i)}} deg_{r} ω;

thus D is isobaric and, by (1), with weight

wt(a_{1}^{μ_{1}} ⋯ a_{r}^{μ_{r}}) = μ_{r} wt(a_{r}) = D_{r}d_{r} = D.

�

Lemma 41.3.5 (Macaulay). Any other determinant D′ of the above matrix has a common factor with D which is homogeneous of degree D_{r} in the coefficients of F_{r}.

Proof. Let us denote H_{j}, 1 ≤ j ≤ \binom{d+r−1}{r−1} the polynomials corresponding to the chosen rows of the above matrix; according Theorem 41.2.3, any arbitrarily 𝕂-linear combination

∑_{j} α_{j}H_{j} = ∑_{i=1}^{r} A_{i}F_{i} ∈ Span_{𝕂}(W_{d})

has a unique representation

∑_{i=1}^{r} A_{i}F_{i} = ∑_{i=1}^{r} Q_{i}F_{i}, Q_{i} ∈ Span_{𝕂}(W_{d}^{(i)}).

There is therefore a matrix M ∈ GL(\binom{d+r−1}{r−1}, 𝕂) such that D det(M) = D′.

We can now compute each Q_{i} ∈ Span_{𝕂}(W_{d}^{(i)}) and therefore the matrix M, by the following recursive procedure:

• compute the polynomials (whose existence is implied by Theorem 41.2.3) Y_{i}^{(r)} ∈ Span_{𝕂} (W_{d−d_{r}}^{(i)}), i ≤ r, such that A_{r} = ∑_{i=1}^{r−1} Y_{i}^{(r)}F_{i} + Y_{r}^{(r)}; since

∑_{i=1}^{r} A_{i}F_{i} = ∑_{i=1}^{r−1} A_{i}F_{i} + A_{r}F_{r}
= ∑_{i=1}^{r−1} A_{i}F_{i} + ∑_{i=1}^{r−1} Y_{i}^{(r)}F_{i}F_{r} + Y_{r}^{(r)}F_{r}
= ∑_{i=1}^{r−1}(A_{i} + Y_{i}^{(r)}F_{r})F_{i} + Y_{r}^{(r)}F_{r}

we can set Q_{r} := Y_{r}^{(r)} and reduce the problem of solving the equation

∑_{i=1}^{r} A_{i}F_{i} = ∑_{i=1}^{r} Q_{i}F_{i}

to the one of solving ∑_{i=1}^{r−1}(A_{i} + Y_{i}^{(r)}F_{r})F_{i} = ∑_{i=1}^{r−1} Q_{i}F_{i};

• Q_{r−1} is then obtained by computing the polynomials (whose existence is implied by Theorem 41.2.3) Y_{i}^{(r−1)} ∈ Span_{𝕂} (W_{d−d_{r−1}}^{(i)}), i ≤ r − 1, such that

A_{r−1} + Y_{r−1}^{(r)}F_{r} = ∑_{i=1}^{r−2} Y_{i}^{(r−1)}F_{i} + Y_{r−1}^{(r−1)},

allowing to obtain, setting Q_{r−1} := Y_{r−1}^{(r−1)},

∑_{i=1}^{r} A_{i}F_{i}

= Q_{r}F_{r} + ∑_{i=1}^{r−1}(A_{i} + Y_{i}^{(r)}F_{r})F_{i}
= Q_{r}F_{r} + ∑_{i=1}^{r−2}(A_{i} + Y_{i}^{(r)}F_{r})F_{i} + ∑_{i=1}^{r−2} Y_{i}^{(r−1)}F_{i}F_{r−1} + Y_{r−1}^{(r−1)}F_{r−1}
= ∑_{i=1}^{r−2}(A_{i} + Y_{i}^{(r)}F_{r} + Y_{i}^{(r−1)}F_{r−1})F_{i} + ∑_{i=r−1}^{r} Q_{i}F_{i};

• inductively we assume to have

∑_{i=1}^{r} A_{i}F_{i} = ∑_{i=1}^{r−j} (A_{i} + ∑_{l=r−j+1}^{r} Y_{i}^{(l)}F_{l}) F_{i} + ∑_{i=r−j+1}^{r} Q_{i}F_{i}

and we compute the polynomials Y_{i}^{(r−j)} ∈ Span_{𝕂} (W_{d−d_{r−j}}^{(i)}), i ≤ r − j, such that

A_{r−j} + ∑_{l=r−j+1}^{r} Y_{r−j}^{(l)}F_{l} = ∑_{i=1}^{r−j−1} Y_{i}^{(r−j)}F_{i} + Y_{r−j}^{(r−j)},

so that, setting Q_{r−j} := Y_{r−j}^{(r−j)}, we obtain

∑_{i=1}^{r} A_{i}F_{i}

= ∑_{i=1}^{r−j} (A_{i} + ∑_{l=r−j+1}^{r} Y_{i}^{(l)}F_{l}) F_{i} + ∑_{i=r−j+1}^{r} Q_{i}F_{i}

= ∑_{i=1}^{r−j−1} (A_{i} + ∑_{l=r−j+1}^{r} Y_{i}^{(l)}F_{l}) F_{i}

+ ∑_{i=1}^{r−j−1} Y_{i}^{(r−j)}F_{i}F_{r−j} + Y_{r−j}^{(r−j)}F_{r−j} + ∑_{i=r−j+1}^{r} Q_{i}F_{i}

= ∑_{i=1}^{r−j−1} (A_{i} + ∑_{l=r−j}^{r} Y_{i}^{(l)}F_{l}) F_{i} + ∑_{i=r−j}^{r} Q_{i}F_{i}

• und so weiter.

The point is that in this procedure all the polynomials Y_{i}^{(l)} are completely independent from the coefficients a_{r,τ} of F_{r} and the same is true for det(M) ∈

𝕂 whence the claim.  �

The choice of the determinant D being dependent on an ordering on the variables, we can in fact choose r! diffierent determinants D_{π}, π ∈ S_{r}, each satisfying (cf. Remark 41.2.4) a proper reformulation of Theorem 41.2.3.

In particular

Corollary 41.3.6. It holds:

(1) for each i ≤ r and each π ∈ S_{r} such that π(r) = i the determinant D_{π} is homogeneous of degree D_{i} in the coefficients of F_{i};

(2) any other determinant D′ has a common factor with such determinant which is homogeneous of degree D_{i} in the coefficients of F_{i}.

(3) R is homogeneous of degree D_{i} = μ_{i} in the coefficients of F_{i} for each i.

Example 41.3.7. In Example 41.3.1 if we consider the permutation (132) the

construction would return Figure 41.2  �

Definition 41.3.8. The element A ∈ 𝔻 such that D = AR is called the extraneous factor of D.

Fig. 41.2. Macaulay’s Matrix (2)

x^{3} x^{2}y x^{2}z xy^{2} xyz xz^{2} y^{3} y^{2}z yz^{2} z^{3}
x^{2}F_{3} x x X  0  0  0 0 0  0 0
xyF_{3} 0 x  0  x X  0 0 0  0 0
xzF_{3} 0 0  x  0  x X 0 0  0 0
y^{2}F_{3} 0 0  0  x  0  0 x X 0 0
yzF_{3} 0 0  0  0  x  0 0 x X 0
z^{2}F_{3} 0 0  0  0  0  x 0 0  x X
xF_{1} X x  x  x  x  x 0 0  0 0
yF_{1}  0 X  0  x  x  0 x x  x 0
zF_{1}  0 0  X  0  x  x 0 x  x x
xF_{2} x x  x X x  x 0 0  0 0
yF_{2}  0 x  0  x  x  0 X x  x 0
zF_{2}  0 0  x  0  x  x 0 X  x x

Denoting χ_{ν} : 𝔻[Z_{1}, …, Z_{r}] → 𝔻[Z_{1}, …, Z_{ν−1}] the evaluation

F (Z_{1}, …, Z_{r}) ↦ χ_{ν}(F) = F (Z_{1}, …, Z_{ν−1}, 0, …, 0),

we have:

Corollary 41.3.9. The following holds

(1) c(R, a_{1}^{D_{1}} ⋯ a_{r}^{D_{r}}) = ±1;

(2) R is homogeneous of degree D_{i} in the coefficients of F_{i} for each i;

(3) R is isobaric of degree D;

(4) A is isobaric of degree 0;

(5) c(A, a_{i,τ}) = 0 for each τ ∈ W : Z_{r} | τ;

(6) A is independent of the coefficients a_{r,τ} of F_{r};

(7) A depends only on the coefficients of the generic polynomials

χ_{r}(F_{1}), …, χ_{r}(F_{r−1}) ∈ ℤ[a_{i,τ} , 1 ≤ i < r, τ ∈ W_{d_{i}}][Z_{1}, …, Z_{r−1}];

(8) Ξ_{f} (c_{i}) = Ξ_{f} (a_{i,Z_{r}^{d}}) = c(f_{i}, Z_{r}^{d}) = 0 for each i ⟹ Ξ_{f} (R) = 0.

Proof. (1) follows from Proposition 41.3.4(1) and from the equality D_{i} = μ_{i} for each i; (2) is a direct consequence of Corollary 41.3.6.

Ad (3): each factor of an isobaric function is isobaric too; the weight of R is wt(a_{1}^{D_{1}} ⋯ a_{r}^{D_{r}}) = D.

(4) is a direct corollary of (3) and (5) of (4); (6) is the statement at the end of the proof of Lemma 41.3.5 and (7) resumes (5-6).

(8) follows from Proposition 41.3.4(2), which implies Ξ_{f} (D) = 0 and on

(5) which implies that A is independent of each c_{i} = a_{i,Z_{r}^{d}}.  �

41.4 The Extraneous Factor

In order to give an explicit and effective representation of the extraneous factor A of D, Macaulay needed a deeper analysis and a convenient notation, starting from Bézout’s formula (Theorem 41.2.3). Thus considering, for each ν ≤ r and each δ ≥ min(d_{1}, …, d_{ν}), the submatrix obtained by selecting the columns indexed by the terms in

W_{δ} \ W_{δ}^{(ν+1)} = {τZ_{1}^{d_{1}}, τ ∈ W_{δ}^{(1)}} ⊔ ⋯ ⊔ {τZ_{ν}^{d_{ν}} , τ ∈ W_{δ}^{(ν)}}

and the rows indexed by the elements

ωF_{i} = ∑_{τ∈W_{d}} c(ωF_{i}, τ)τ ∈ B := {ωF_{i} : ω ∈ W_{δ}^{(i)}, 1 ≤ i ≤ ν}.

Macaulay denotes D(ν, δ) its determinant and^{7}

R(ν, δ) := gcd(D_{π}(ν, δ) : π ∈ S_{ν}}.

He also sets D(ν, δ) = R(ν, δ) = 1 for each δ < min(d_{1}, …, d_{ν}).

Remark 41.4.1. The following conditions are equivalent:

(1) Ξ_{f} (D(ν, δ)) = 0

(2) ∑_{i=1}^{ν} Q_{i}χ_{ν}(f_{i}) = Q_{ν+1} for suitable Q_{i} = ∑_{ω∈W_{δ}^{(i)}} c(Q_{i}, ω)ω ∈ Q, i ≤ ν + 1.

Moreover, it is sufficient to repeat the same argument which led to the proof of Corollary 41.3.6, in order to obtain that R(ν, δ) is homogeneous, for each i ≤ ν, in the coefficients of F_{i} with degree

#{Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{i}} : a_{j} < d_{j} for each j ≠ i}.

�

This notation allows to state

Theorem 41.4.2 (Macaulay). It holds

|\frac{D(r, δ)}{R(r, δ)}| = | ∏_{j=0}^{d_{r}−1} \frac{D(r − 1, δ − j)}{R(r − 1, δ − j)}| · | ∏_{j=d_{r}}^{δ−1} D(r − 1, δ − j)| .

only.

7 In this construction the value ν fixes the precise set {Z_{1}, ⋯, Z_{ν}} of the first ν

variables; therefore, in this setting Remark 41.2.4 is appliable to those variables

Proof. R(r, δ) is a factor of D(r, δ) and (Corollary 41.3.9(6)) the other factors are independent of the coefficients of F_{r}.

Let us consider an arbitrary combination

∑_{i=1}^{r} Q_{i}F_{i} = 0, Q_{i} = ∑_{ω∈W_{δ}^{(i)}} x(Q_{i}, ω)ω, 1 ≤ i ≤ r

in terms of the unknowns x(Q_{i}, ω), where

W_{δ}^{(i)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{i}} : a_{j} < d_{j} for each j < i}

and denote a := a_{r,Z_{r}^{dr}} and w := #W_{δ}^{(r)}.

The element a appears w times in the matrix, namely in the positions satisfying c(ωF_{r}, ωZ_{r}^{d_{r}}) = a = a_{r,Z_{r}^{dr}} where ω runs in the elements of W_{δ}^{(r)}; more precisely, for each such ω, a appears in the position corresponding to the column indexed by ωZ_{r}^{d_{r}} and the row representing ωF_{r}.

The columns where a does not appear are those indexed by the terms

{τZ_{1}^{d_{1}}, τ ∈ W_{δ}^{(1)}} ⊔ ⋯ ⊔ {τZ_{r−1}^{d_{r−1}}, τ ∈ W_{δ}^{(r−1)}} .

Hence the coefficient of a^{w} in the expansion of D(r, δ) is the determinant whose vanishing is the condition that the identity

∑_{i=1}^{r−1} Q_{i}F_{i} = Q_{r} Q_{i} = ∑_{ω∈W_{δ}^{(i)}} x(Q_{i}, ω)ω, 1 ≤ i ≤ r

can be satisfied.

In order to evaluate such determinant, assume the identity is satisfied and set Z_{r} = 0 obtaining the identity

∑_{i=1}^{r−1} χ_{r}(Q_{i})χ_{r}(F_{i}) = χ_{r}(Q_{r}).

Therefore, for each f := {f_{1}, …, f_{r}}, either

• such identity is non-trivially satisfied, i.e. Ξ_{f} (D(r − 1, δ)) = 0, or

• χ_{r}(Q_{1}) = ⋯ = χ_{r}(Q_{r}) = 0, which means that Z_{r} | Q_{i} for each i.

In the latter case we obtain a similar identity

∑_{i=1}^{r−1} Q′_{i}f_{i} = Q′_{r}, Q′_{i} = ∑_{ω∈W_{δ−1}^{(i)}} x(Q_{i}, Z_{r}ω)ω, 1 ≤ i ≤ r.

Repeating the same argument we obtain that either Ξ_{f} (D(r−1, δ−1)) = 0, or there is an identity

∑_{i=1}^{r−1} Q′′_{i}f_{i} = Q′′_{r}, Q′′_{i} = ∑_{ω∈W_{δ−2}^{(i)}} x(Q_{i}, Z_{r}^{2}ω)ω, 1 ≤ i ≤ r.

Since we have only the limitation

max {a_{r} : Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ ∪_{i=1}^{r−1}W_{δ}^{(i)}} = δ

by iteration we deduce that the sought determinant is ∏_{j=0}^{δ−1} D(r − 1, δ − j) so that

D(r, δ) = (∏_{j=0}^{δ−1} D(r − 1, δ − j)) a^{w} + ⋯

Let us therefore now evaluate the coefficient of a^{w} in R(r, δ); to do so we consider another permutation, say the cyclic one, π(i) ≡ i − 1 (mod r) and the corresponding identity

Q_{1}F_{r} + Q_{2}F_{1} + ⋯ + Q_{r}F_{r−1} = 0, Q_{i} = ∑_{ω∈W_{πδ}^{(i)}} x(Q_{i}, ω)ω, 1 ≤ i ≤ r

where W_{πδ}^{(i)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{π(i)}} : a_{π(j)} < d_{π(j)} for each j < i} and, in particular,

W_{πδ}^{(1)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{r}} : a_{r} < d_{r}}.

The variable a := a_{r,Z_{r}^{dr}} appears s := #W_{πδ}^{(1)} times in the positions satisfying c(ωF_{r}, ωZ_{r}^{d_{r}}) = a and corresponding to the column indexed by ωZ_{r}^{d_{r}} and the row representing ωF_{r}, where ω runs in the elements of W_{πδ}^{(1)}.

The remaining columns are those indexed by the terms

{τZ_{1}^{d_{1}}, τ ∈ W_{πδ}^{(2)}} ⊔ ⋯ ⊔ {τZ_{r−1}^{d_{r−1}}, τ ∈ W_{πδ}^{(r−1)}}

and the coefficient of a^{s} in the expansion of D_{π}(r, δ) is the determinant whose vanishing is the condition that the identity

∑_{i=1}^{n−1} Q_{i+1}F_{i} = Q_{1} Q_{i} = ∑_{ω∈W_{πδ}^{(i)}} x(Q_{i}, ω)ω, 1 ≤ i ≤ r

can be satisfied.

We can therefore reapply the same argument as above, setting Z_{r} = 0 and obtaining that, for each f := {f_{1}, …, f_{r}}, either Ξ_{f} (D_{π}(r − 1, δ)) = 0 or each Q_{i} is divisible by Z_{r}; however, since we have the stricter limitation

max {a_{r} : Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ ∪_{i=1}^{r−1}W_{δ}^{(i)}} = d_{r}

we obtain only

D_{π}(r, δ) = ( ∏_{j=0}^{d_{r}−1} D_{π}(r − 1, δ − j)) a^{s} + · · · .

Since the same argument can be applied for each permutation π′, π′(1) = r, we obtain

R(r, δ) = ( ∏_{j=0}^{d_{r}−1} R(r − 1, δ − j)) a^{s} + · · · .

Since R(r, δ) is a factor of D(r, δ) and the other factors are independent of

the coefficients of F_{r} we thus obtain the claim.  �

Definition 41.4.3. A term Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W satisfying

a_{j} < d_{j} for each j ∈ {i_{1}, …, i_{h}} ⊂ {1, …, r}

is said to be reduced in {Z_{i_{1}}, …, Z_{i_{h}}}.

Let S ⊂ W be the semigroup ideal generated by {Z_{i}^{d_{i}}, 1 ≤ i ≤ r}; S_{⋆} ⊂ W be the semigroup ideal generated by {Z_{i}^{d_{i}}Z_{j}^{d_{j}}, 1 ≤ i < j ≤ r}; W_{⋆} ⊂ W be the set of terms which is divisible by a single term Z_{i}^{d_{i}}; W_{⋆δ}^{(i)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ}^{(i)} : a_{j} < d_{j} for each j > i} U_{δ}^{(i)} := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ}^{(i)} : exists h > i : a_{h} ≥ d_{h}}. and set S_{δ} := S ∩ W_{δ}, W_{⋆δ} := W_{⋆} ∩ W_{δ} and S_{⋆δ} := S_{⋆} ∩ W_{δ}. Then

Lemma 41.4.4. It holds

(1) W \ S = ⋃_{δ} W_{δ}^{(r+1)} is the set of all terms reduced in {Z_{i}, 1 ≤ i ≤ r};

(2) W = W_{⋆} ⊔ S_{⋆} ⊔ ∪_{δ}W_{δ}^{(r+1)};

(3) W_{δ} \ W_{δ}^{(r+1)} = W_{⋆δ} ⊔ S_{⋆δ};

(4) W_{δ} = W_{⋆δ} ⊔ S_{⋆δ}, for each δ ≥ d;

(5) W_{⋆} = ⋃_{i=1}^{r}{Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W \ ⋃_{δ} W_{δ}^{(r+1)} : a_{j} < d_{j} for each j ≠ i};

(6) W_{⋆} is the set of all terms which are reduced in {Z_{j}, j ≠ i} for some i but are not reduced in {Z_{1}, …, Z_{r}};

(7) W_{⋆δ}^{(i)} = {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{i}} : a_{j} < d_{j} for each j ≠ i} is the set of terms of degree δ − d_{i} which are reduced in {Z_{1}, …, Z_{i−1}, Z_{i+1}, …, Z_{r}};

(8) \overline{W}_{δ} := W_{⋆δ} ∪ W_{δ}^{(r+1)} = ⋃_{i} W_{⋆δ}^{(i)}; W_{δ}^{(r+1)} = ⋂_{i} W_{⋆δ}^{(i)};

(9) W_{δ}^{(i)} = W_{⋆δ}^{(i)} ⊔ U_{δ}^{(i)};

(10) U_{δ}^{(i)} = {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{i}} : a_{j} < d_{j}∀j < i, ∃h > i : a_{h} ≥ d_{h}};

(11) W_{⋆δ}^{(r)} = W_{δ}^{(r)};

(12) for each τ ∈ W_{⋆δ} there is i ≤ r and ω ∈ W_{⋆δ}^{(i)} such that c(ωF_{i}, τ) = a_{i,Z_{i}^{d_{i}}}.

(13) for each i ≤ r and ω ∈ W_{⋆δ}^{(i)} : c(ωF_{i}, τ) = a_{i,Z_{i}^{d_{i}}} ⟹ τ = ωZ_{i}^{d_{i}} ∈ W_{⋆δ}.

�

Definition 41.4.5. The extraneous factor of D(r, δ), A(r, δ), is the determinant of the minor of D(r, δ) obtained removing the columns indexed by the terms in W_{⋆δ} and the rows indexed by the polynomials which contains the elements a_{i,Z_{i}^{d_{i}}} , 1 ≤ i ≤ r, in the omitted columns, id est the rows indexed by the set

{ωF_{i} : ω ∈ W_{⋆d}^{(i)}, 1 ≤ i ≤ r}.

Alternatively the surviving columns are the ones indexed by the terms τ ∈ S_{⋆δ} and the surviving rows are the ones related to the elements in

{ωF_{i} : ω ∈ U_{d}^{(i)}, 1 ≤ i ≤ r}.

�

Example 41.4.6. In Figures 41.1 and 41.2 the elements of the extraneous factor are represented · .

Note that in Figures 41.2 variables and polynomials are ordered as z, x, y

(respectivley F_{3}, F_{1}, F_{2}).  �

Remark 41.4.7. Since, in the construction of D(ν, δ) and R(ν, δ), the value ν fixes the precise set {Z_{1}, ⋯, Z_{ν}} of the first ν variables, the definition of

extraneous factor can be naturally extended to define A(ν, δ).  �

Theorem 41.4.8 (Macaulay). It holds:

(1) |A(r, δ)| = |∏_{j=0}^{d_{r}−1} A(r − 1, δ − j)| · |∏_{j=d_{r}}^{δ−1} D(r − 1, δ − j)| .

(2) D(2, δ) = A(2, δ)R(2, δ);

(3) D(r, δ) = A(r, δ)R(r, δ).

Proof. Since (2) requires just a trivial verification and allows to deduce (3) from (1), we just need to prove (1).

The vanishing of A(r, δ) is the condition that the identity

∑_{i=1}^{r−1} Q_{i}F_{i} = Q_{r}, Q_{i} = ∑_{ω∈U_{δ}^{(i)}} x(Q_{i}, ω)ω, 1 ≤ i < r, Q_{r} = ∑_{ω∈\overline{W}_{δ}} x(Q_{r}, ω)ω,

can be solved in terms of the unknowns x(Q_{i}, ω).

The number of linear equations and unknown are equal and A(n, δ) is not zero since, for the ansatz Ξ_{Z}(F_{i}) := Z_{i}^{d_{i}}, in the polynomial ∑_{i=1}^{r−1} Q_{i}Z_{i}^{d_{i}} + Q_{r} each term in W_{δ} occurs once and once only.

Assume, again, that the identity is satisfied, set Z_{r} = 0 obtaining the identity

∑_{i=1}^{r−1} χ_{r}(Q_{i})χ_{r}(F_{i}) = χ_{r}(Q_{r}),

and reapply the same argument as in Theorems 41.4.2, obtaining, for each f := {f_{1}, …, f_{r}}, that either Ξ_{f} (A(r −1, δ)) = 0 or each Q_{i} is divisible by Z_{r}; repeating the same argument we can obtain that either Ξ_{f} (A(r − 1, δ − i)) = 0, 0 ≤ i < d_{r} , or there is an identity ∑_{i=1}^{r−1} Q′′_{i}f_{i} = Q′′ where

Q′′_{i} = ∑_{ω∈𝔘_{δ−dr}^{(i)}} x(Q_{i}, Z_{r}^{d_{r}}ω)ω, 1 ≤ i < r, Q′′ = ∑_{ω∈𝔚_{δ−dr}} x(Q_{i}, Z_{r}^{d_{r}}ω)ω,

𝔘_{δ−d_{r}}^{(i)} := \{ω : Z_{r}^{d_{r}}ω ∈ U_{δ}^{(i)}\} and 𝔚_{δ−d_{r}} := \{ω : Z_{r}^{d_{r}}ω ∈ \overline{W}_{δ}\} = W_{δ−d_{r}}^{(r)}. We have a similar relation

𝔘_{δ−d_{r}}^{(i)} := {ω : Z_{r}^{d_{r}}ω ∈ U_{δ}^{(i)}} = W_{δ−d_{r}}^{(i)}

also for i < r since

Z_{r}^{d_{r}}ω ∈ W_{⋆δ}^{(i)} = {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{δ−d_{i}} : a_{j} < d_{j} for each j ≠ i} ⟹ i = r

whence

𝔘_{δ−d_{r}}^{(i)} = {ω : Z_{r}^{d_{r}}ω ∈ U_{δ}^{(i)}}
= {ω : Z_{r}^{d_{r}}ω ∈ W_{δ}^{(i)} \ W_{⋆δ}^{(i)}}
= {ω : Z_{r}^{d_{r}}ω ∈ W_{δ}^{(i)}}
= W_{δ−d_{r}}^{(i)}.

Thus, we can conclude that either Ξ_{f} (A(r − 1, δ − i)) = 0, 0 ≤ i < d_{r} , or there is an identity

∑_{i=1}^{r−1} Q′′_{i}f_{i} = Q′′_{r}, Q′′_{i} = ∑_{ω∈W_{δ−dr}^{(i)}} x(Q_{i}, X_{n}^{d_{n}}ω)ω, 1 ≤ i ≤ r,

id est (by Theorems 41.4.2) ∏_{j=d_{n}}^{δ−1} Ξ_{f} (D(n − 1, δ − j)) = 0.  �

Corollary 41.4.9. The extraneous factor A of D = D(n, d) satisfying A =

\frac{D}{R} = \frac{D(n,d)}{R(n,d)} is A := A(n, d).  �

41.5 Macaulay’s Resultant

Remark 41.5.1. If the resultant Res(f_{1}, …, f_{r}) vanish, then f = {f_{1}, …, f_{r}} have a common root α ∈ ℙ^{r−1}(K) and all polynomials in

Ξ_{f}(B) := {ωf_{i} : ω ∈ W_{d}^{(i)}, 1 ≤ i ≤ n}

vanish when evaluated at such root; thus, setting x_{τ} := τ(α) for each τ ∈ W, (x_{τ} : τ ∈ W_{d}) is a common root of the linear equations

∑_{τ∈W_{d}} x_{τω}c(f_{i}, τ) = 0, ω ∈ W_{d}^{(i)}, 1 ≤ i ≤ n

and Ξ_{f}(D_{π}) = 0 for each π ∈ S_{r}.

Thus Res(d_{1}, …, d_{r}) divides each D_{π}, π ∈ S_{r} and hence divides R; since both R (Corollary 41.3.9) and Res(d_{1}, …, d_{r}) (Fact 41.1.2(2)) are isobaric of weigt D, in principle, we can conclude that R is the sought-after resultent. However, since we have not given a complete proof of Fact 41.1.2(2), we prefer to explicitly proof that R is the resultent, and deduce Fact 41.1.2(2)

from Theorem 41.5.3 below.  �

Lemma 41.5.2 (Macaulay). The coefficients of a generic member of J_{d−1} satisfy one and only one identical linear relation.^{8}

Proof. We need to prove that dim_{𝕂}(J_{d−1}) = #W_{d−1} − 1.

In fact the equation

∑_{i=1}^{r} A_{i}F_{i} = ∑_{i=1}^{r} Q_{i}F_{i}, Q_{i} ∈ Span_{𝕂}(W_{d−1}^{(i)})

can be solved by the method used in Lemma 41.3.5 for arbitary given polynomials A_{i}; thus dim_{𝕂}(J_{d−1}) is less or equal on the number of coefficients in the expression ∑_{i=1}^{r} Q_{i}Z_{i}^{d_{i}} which is #W_{d−1} − 1 since each term in #W_{d−1} except Ω := ∏_{i=1}^{r} Z_{i}^{d_{i}−1} occurs once and only once in that expression.

In order to prove that this equality is strict, it is sufficient to show that that it is satisfied by at least a specific ansatz . Macaulay consider’s the ansatz Ξ_{1}(F_{i}) := f_{i}, 1 ≤ i ≤ r where

f_{i} := (Z_{i} − Z_{i+1})Z_{i}^{d_{i}−1}, 1 ≤ i < r, f_{r} := (Z_{r} − Z_{1})Z_{r}^{d_{r}−1};

clearly Ξ_{1}(R) = 0 since the system f_{1} = ⋯ f_{r} = 0 has the common root (1, 1, …, 1).

8 Both this result and the Theorem below requires d ≥ 2 id est the existence of at

least a non-linear polynomial.

On the other side, if d_{i} = 1 for each i so that d = 1, this Lemma is empty but the Theorem below claims that the determinant of a system of r linear equations in r variables vanishes if and only if the system has a common root.

In order to prove that dim_{K}(Ξ_{1}(J)_{d−1}) = #W_{d−1} − 1, Macaulay shows that for each term τ := Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}}, τ−Ω ∈ (f_{1}, …, f_{n}) = Ξ_{1}(J) by proposing an interesting rewriting procedure which is worthwhile to quote; given τ, set ι := 1 and repeatedly perform the following transformation:

• if a_{ι} ≥ d_{ι} set τ := Z_{1}^{a_{1}} ⋯ Z_{ι−1}^{a_{ι−1}}Z_{ι}^{d_{ι}−1}Z_{ι+1}^{a_{ι+1}+a_{ι}−d_{ι}+1} ⋯ Z_{r}^{a_{r}} which is equivalent to transform τ to

τ − (Z_{ι}^{a_{ι}} − Z_{ι+1}^{a_{ι}−d_{ι}+1})\frac{τ}{Z_{ι}^{a_{ι}}}

= τ − (Z_{ι}^{a_{ι}−d_{ι}+1} − Z_{ι+1}^{a_{ι}−d_{ι}+1})Z_{ι}^{d_{ι}−1} \frac{τ}{Z_{ι}^{a_{ι}}}
= τ − \frac{Z_{ι}^{a_{ι}−d_{ι}+1} − Z_{ι+1}^{a_{ι}−d_{ι}+1}}{Z_{ι} − Z_{ι+1}} \frac{τ}{Z_{ι}^{a_{ι}}} (Z_{ι} − Z_{ι+1})Z_{ι}^{d_{ι}−1}
= τ − \frac{Z_{ι}^{a_{ι}−d_{ι}+1} − Z_{ι+1}^{a_{ι}−d_{ι}+1}}{Z_{ι} − Z_{ι+1}} \frac{τ}{Z_{ι}^{a_{ι}}} Ξ_{1}(f_{ι})

• ι := ι + 1 mod r

going round the cycle^{9} Z_{1}, Z_{2}, …, Z_{r}, Z_{1} as many time as needed until we

obtain the term Ω.  �

Theorem 41.5.3 (Macaulay). R = Res(F_{1}, …, F_{r}).

Proof. We have (Proposition 41.3.4(3)) ARZ_{r}^{d} ∈ J; setting Z_{r} := 1 and applying Kronecker substitution which changes c_{i} to c_{i} − F_{i}, 1 ≤ i ≤ r, then A is not changed (as a consequence of Proposition 41.3.4(5)) while R is changed in R − ∑_{i=1}^{r} A_{i}F_{i}; as a consequence R ∈ (F_{1}, …, F_{r}, Z_{r} − 1) .

Hence Ξ_{f} (R) = 0 if the equations f_{1} = … = f_{n} = 0 have a proper solution (z_{1}, …, z_{r−1}, 1) ∈ ℙ^{r−1}(K).

Let us assume that Ξ_{f} (R) = 0 is a relation among the coefficients of f_{1}, …, f_{r} so that there are less then #W_{d} linearly independent members in Ξ_{f} (B) := {ωf_{i} : ω ∈ W_{δ}^{(i)}, 1 ≤ i ≤ r}.

Hence the coefficients x_{τ} of the generic element

∑_{τ∈W_{d}} x_{τ} τ ∈ Ξ_{f} (J)_{d} = (f_{1}, …, f_{n})_{d}

must satisfy a linear relation ∑_{τ∈W_{d}} x_{τ} c_{τ} = 0.

Moreover, by Lemma 41.5.2, also the generic element

9 For r = 4 d_{i} = 4 and d − 1 = 12 we have e.g.

\begin{aligned}Z_{1}^{7}Z_{4}^{5} & → & Z_{1}^{3}Z_{2}^{4}Z_{4}^{5} & → & Z_{1}^{3}Z_{2}^{3}Z_{3}Z_{4}^{5} & → \\ Z_{1}^{5}Z_{2}^{3}Z_{3}Z_{4}^{3} & → & Z_{1}^{3}Z_{2}^{5}Z_{3}Z_{4}^{3} & → & Z_{1}^{3}Z_{2}^{3}Z_{3}^{3}Z_{4}^{3} \end{aligned}

f := ∑_{υ∈W_{d−1}} x_{υ}υ ∈ Ξ_{f} (J)_{d−1} = (f_{1}, …, f_{n})_{d−1}

satisfies a linear relation ∑_{υ∈W_{d−1}} x_{υ}c_{υ} = 0.

Since each Z_{i}f = ∑_{υ∈W_{d−1}} x_{υ}Z_{i}υ ∈ (f_{1}, …, f_{n})_{d} the unknowns x_{υ} must satisfy the r equations

∑_{υ∈W_{d−1}} x_{υ}c_{υZ_{i}} = 0

which are necessarily equivalent so that for each υ ∈ W_{d−1} the continued ratio c_{υZ_{1}} : c_{υZ_{2}} : ⋯: c_{υZ_{r}} is the same; denoting it α_{1} : α_{2} : ⋯: α_{r} it follows that for each τ := Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} ∈ W_{d} c_{τ} is proportional to α_{1}^{a_{1}} ⋯ α_{r}^{a_{r}}

id est α := (α_{1}, …, α_{r}) ∈ ℙ^{r−1}(K) satisfies f_{1}(α) = … = f_{n}(α) = 0.  �

Corollary 41.5.4. With the notation of Corollary 41.3.9 and denoting

R_{ρ} := Res (χ_{ρ}(F_{1}), …, χ_{ρ}(F_{ρ−1}))

also the following holds

(9) c(R, a_{1}^{D_{1}} ⋯ a_{r−1}^{D_{r−1}}) = a_{r}^{D_{r}};

(10) c(R, a_{r}^{D_{r}}) = R_{r}^{d_{r}};

(11) c(R, a_{ρ}^{D_{ρ}} ⋯ a_{r}^{D_{r}}) = R_{ρ}^{d_{ρ}···d_{r}}.

Proof. (9) is obvious and (11) is a repreated applications of (10).

Ad (10): if we consider an ansatz Ξ : Ξ(a_{r,τ}) = 0 for each τ ≠ Z_{r}^{d_{r}}, then Ξ(F_{r}) = Z_{r}^{d_{r}} and Ξ(D) = a_{n}^{D_{n}} R̄ where R̄ is the sub-determinant whose rows correspond to the basis elements {ωF_{i} : ω ∈ W_{δ}^{(i)}, 1 ≤ i < r} and whose columns are labelled by the terms

{τZ_{1}^{d_{1}}, τ ∈ W_{δ}^{(1)}} ⊔ ⋯ ⊔ {τZ_{r−1}^{d_{r−1}}, τ ∈ W_{δ}^{(r−1)}};

the vanishing of such determinant under an ansatz Ξ : Ξ(F_{r}) = Z_{r}^{d_{r}} is a condition that the identity

∑_{i=1}^{r−1} Q_{i}Ξ(F_{i}) = Q_{r}Z_{r}^{d_{r}} Q_{i} = ∑_{ω∈W_{δ}^{(i)}} x(Q_{i}, ω)ω, 1 ≤ i ≤ r

can be satisfied.

In order to evaluate such determinant, assume the identity is satisfied and set Z_{r} = 0 obtaining the identity

∑_{i=1}^{r−1} χ_{r}(Q_{i})χ_{r}(F_{i}) = 0;

thus R̄ is necessarily a multiple of R_{r}^{d_{r}}; evaluating the weight gives that the

multiplicity is d_{r}.  �

Proposition 41.5.5. It holds

(1) Res(F_{1}, …, F_{r−1}, F′_{r}F′′_{r}) = Res(F_{1}, …, F_{r−1}, F′_{r}) Res(F_{1}, …, F_{r−1}, F′′_{r});

(2) Res(F_{1}, F_{2}, …, F_{r−1}, F_{r}) is irreducible.

Proof.

(1) Since for each ansatz

Res(f_{1}, …, f_{r−1}, f′_{r}) Res(f_{1}, …, f_{r−1}, f′′_{r})

vanishes if and only if f_{1} = f_{2} = ⋯ = f′_{r}f′′_{r} = 0 have a common root, we obtain

Res(F_{1}, …, F_{r−1}, F′_{r}F′′_{r}) | Res(F_{1}, …, F_{r−1}, F′_{r}) Res(F_{1}, …, F_{r−1}, F′′_{r});

equality is then obtained since both are isobaric of the same weight.

(2) By contradiction let d_{r} be the least value for which Res(d_{1}, …, d_{r}) = Res(F_{1}, …, F_{r}) has a non trivial factorization Res(d_{1}, …, d_{r}) = R_{1}R_{2}. Choose any two positive values d′_{r} and d′′_{r} such that d_{r} = d′_{r} + d′′_{r}. By the minimality of d_{r} both Res(d_{1}, …, d′_{r}) = Res(F_{1}, …, F′_{r}) and Res(d_{1}, …, d′′_{r}) = Res(F_{1}, …, F′′_{r}) are irreducible. Denote a_{τ} := c(τ, F_{r}), a′_{ω} := c(ω, F′_{r}) and a′′_{υ} := c(υ, F′′_{r}) for each τ ∈ W_{d_{r}}, ω ∈ W_{d′_{r}}, υ ∈ W_{d′′_{r}}. From the ansatz

Ξ(a_{τ}) = ∑_{{ω∈T_{d′_{r}} \atop ω|τ}} a′_{ω}a′′_{\frac{τ}{ω}}

which implies Ξ(F_{r}) = F′_{r}F′′_{r} we have, by (1)

Ξ(R_{1})Ξ(R_{2}) = Res(F_{1}, …, F′_{r}F′′_{r}) = Res(F_{1}, …, F′_{r}) Res(F_{1}, …, F′′_{r});

the irreducibility of R_{1}, R_{2}, Res(F_{1}, …, F′_{r}) and Res(F_{1}, …, F′′_{r}) which is a consequence of the minimality of d_{r} implies, say,

Ξ(R_{1}) = Res(F_{1}, …, F′_{r}), Ξ(R_{2}) = Res(F_{1}, …, F′′_{r}).

Since both R_{1} and R_{2} depend on the a_{τ} s, Ξ(R_{1}) = Res(F_{1}, …, F′_{r}) depends

not only on a′_{ω} but also on a′′_{υ}. This clearly leads to a contradiction.  �

Remark 41.5.6 (Lazard). Given h ≥ r ’generic’ forms

F_{1}, …, F_{h} ∈ 𝔻[Z_{1}, …, Z_{r}], deg(F_{i}) := d_{i}, 𝔻 := ℤ[a_{i,τ} , 1 ≤ i ≤ h, τ ∈ W_{d_{i}}],

d_{1} ≥ d_{2} ≥ ⋯ ≥ d_{r} ≥ ⋯ ≥ d_{h}, d = 1 − r + ∑_{i=1}^{r} d_{i}, the construction of Macaulay’s matrix M, discussed in page 69, can be generalized in order to obtain h blocks, the i^{th} block consisting of the \binom{d−d_{i}+r−1}{r−1} rows related to the K-generators ωf_{i}, ω ∈ W_{d−d_{i}}.

Consequently the notion of Macaulay’s resultant (Definition 41.3.2) generalizes naturally to this setting, still being the greatest common divisor of all determinants of the \binom{d+r−1}{r−1}×∑_{i=1}^{h} \binom{d−d_{i}+r−1}{r−1} Macaulay’s matrix. It is clear that such Macaulay’s resultant R := R(F_{1}, …, F_{h}) is the greatest common divisor of the \binom{h}{r} original Macaulay’s resultants R(F_{j_{1}}, F_{j_{2}}, …, F_{j_{r}}) obtained choosing any subset {j_{1}, …, j_{r}} ⊂ {1, …, h} of r indices.

Equally trivially, denoting, for each set of forms f := {f_{1}, …, f_{h}} ∈ P, Ξ_{f} the ansatz Ξ_{f} (F_{i}) = f_{i}, f has a common root if and only if Ξ_{f} (R) = 0

10.  �

Remark 41.5.7. In the non-homogeneous setting^{11}

the resultant of n given non-homogeneous polynomials in n − 1 variables is the resultant of the corresponding homogeneous polynomials of the same degree obtained by introducing a variable x_{0} of homogeneity.

In other words, given h ≥ r non-homogeneous polynomials

f_{1}, …, f_{k} ∈ K[Z_{1}, …, Z_{r}], d_{i} := deg(f_{i})

and introducing the homogenaizing variable Z_{0}, we consider the generic forms

F_{i} ∈ 𝔻[Z_{0}, Z_{1}, …, Z_{r}], deg(F_{i}) := d_{i}

and the ansatz

Ξ : 𝔻[Z_{0}, Z_{1}, …, Z_{r}] → K[Z_{0}][Z_{1}, …, Z_{r}] : Ξ(F_{i}) := Z_{0}^{d}f_{i}(\frac{Z_{1}}{Z_{0}} , …, \frac{Z_{r}}{Z_{0}} );

more precisely, we consider the \binom{d+r}{r} × ∑_{i=1}^{h} \binom{d−d_{i}+r}{r} Macaulay’s matrix whose columns are indexed by the set W(d) of all terms of degree bounded by d and whose rows represent the polynomials

ωZ_{0}^{d}f_{i}(\frac{Z_{1}}{Z_{0}} , …, \frac{Z_{r}}{Z_{0}} ), ω ∈ W(d − d_{i}), 1 ≤ i ≤ h,

the corresponding resultant being an element of K[Z_{0}].  �

10 Macaulay’s statements consider only the case h = r but such result is already

implicit in the introduction of the determinants D(ν, δ) (page 76).

Moreover, the construction of the u-resultants, in the next Section, freely uses these implicit definitions and constructions.

11 F. S. Macaulay, The Algebraic Theory of Modular Systems, Cambridge Univ. Press (1916), pg. 3.

41.6 Macaulay: The u-resultant. 87

41.6 Macaulay: The u-resultant.

Let us consider r ≤ n homogeneous polynomials

f := {f_{1}, …, f_{r}} ⊂ k[x_{1}, …, x_{n}]

of degrees d_{1} ≤ ⋯ ≤ d_{r}.

One can therefore expect that the ideal M := (f_{1}, …, f_{r}) has rank r, in which case Macaulay considers^{12} its extension/contraction ideal

M^{(r)} := Mk(x_{r+1}, …, x_{n})[x_{1}, …, x_{r}] ∩ k[x_{1}, …, x_{n}]

and is aware that, if a ’generic’ change of coordinates has been already performed, each f_{i} ∈ M^{(r)} is homogeneous of degree d_{i} in the variables^{13} x_{1}, …, x_{r} and the assumption on the rank is satisfied if and only if the resultant of the r f_{i}s w.r.t. the r−1 variables x_{1}, …, x_{r−1}, F_{f} ∈ k[x_{r+1}, …, x_{n}][x_{r}] does not vanish, thus granting the existence of a root.

In this context and under these assumptions, adapting the notation of Chapters 31-32 and 39 we set

k[x_{1}, …, x_{n}] = k[Z_{1}, …, Z_{r}, V_{1}, …, V_{d}] = k[x_{r+1}, …, x_{n}][x_{1}, …, x_{r}],

denote^{14} π : k[x_{1}, …, x_{n}] = k[V_{1}, …, V_{d}][Z_{1}, …, Z_{r}] → k[Z_{1}, …, Z_{r}] the projection defined by π(F) = F (Z_{1}, …, Z_{r}, 0, …, 0], for each

F (x_{1}, …, x_{r}, x_{r+1}, …, x_{n}) = F (Z_{1}, …, Z_{r}, V_{1}, …, V_{d}),

K = k(V_{1}, …, V_{d}), R = k[V_{1}, …, V_{d}], and consider

(f_{1}, …, f_{r}) = M^{(r)} ⊂ R[Z_{1}, …, Z_{r}],

remarking that with this new notation we have F_{f} ∈ R[Z_{r}].

We begin by remarking that the polynomials

f̄_{i} := f_{i}(Z_{0}V_{1}, …, Z_{0}V_{d}, Z_{1}, …, Z_{r−1}, Z_{0}Z_{r}) ∈ R[Z_{0}, Z_{1}, …, Z_{r−1}, Z_{r}],

are homogeneous in the variables Z_{1}, …, Z_{r−1}, Z_{0} and that F_{f} ∈ R[Z_{r}] is the resultant Ff̄ w.r.t. Z_{1}, …, Z_{r−1}, Z_{0} of f̄ := {f̄_{1}, …, f̄_{r}}; moreover F_{f} is a homogeneous polynomial in the variables V_{1}, …, V_{d}, Z_{r} of degree D := ∏_{i=1}^{r} d_{i} so that T(Ff̄) = R_{r+1}Z_{r}^{D} where (by the assumption on the rank)

R_{r+1} = Res (π(f_{1}), …, π(f_{r})) ≠ 0.

Instead of solving for one of the unknown variables Z_{i}, we solve for their Liouville substitution

12 Compare the discussion in Section 30.5. 13 x_{r} being chosen as variable of homogeneity. 14 Compare Section 30.5, n. 48.

Z = U_{1}Z_{1} + U_{2}Z_{2} + ⋯ + U_{r}Z_{r}

setting f_{u} := Z − U_{1}Z_{1} − U_{2}Z_{2} − ⋯ − U_{r}Z_{r}, considering the polynomial set f^{(u)} := {f_{1}, …, f_{r}, f_{u}} as a subset of R[U_{1}, …, U_{r}][Z, Z_{1}, …, Z_{r}] and computing their resultant F_{f}^{(u)} ∈ R[U_{1}, …, U_{r}][Z] w.r.t. Z_{1}, …, Z_{r}.

Definition 41.6.1 (Macaulay). F_{f}^{(u)} is called the u-resultant of f.  �

With an argument similar to the one we gave for F_{f}, setting

f̂_{i} := f_{i}(Z_{0}V_{1}, …, Z_{0}V_{d}, Z_{1}, …, Z_{r}), 1 ≤ i ≤ r and f̂_{u} := Z_{0}Z − ∑_{i=1}^{r} U_{i}Z_{i}

and denoting

f̂^{(u)} := {f̂_{1}, …, f̂_{r}, f̂_{u}} ⊂ R[U_{1}, …, U_{r}][Z_{0}, Z_{1}, …, Z_{r}, Z],

we have that F_{f}^{(u)} is the resultant F (u) ; f̂ of f̂^{(u)} w.r.t. Z_{1}, …, Z_{r}, Z_{0} and, being homogeneous in the variables V_{1}, …, V_{d}, Z of degree D := ∏_{i=1}^{r} d_{i}, we have

T(F (u) ; f̂) = R′_{r+1}Z^{D} where R′_{r+1} = Res (π(f_{1}), …, π(f_{r}), π(f_{u})) .

If we consider in the expansion of F (u) ; f̄ the indeterminate coefficient a, representing the coefieient c(Z, f_{u}) of Z in f_{u}, degree considerations allow to deduce that a^{D} | R′_{r+1} and R′_{r+1} = a^{D}R_{r+1} whence R′_{r+1} = R_{r+1} since the ansatz evaluates a as c(Z, f_{u}) = 1.

With the same kind of argument as in Theorem 41.5.3, we can deduce that to each root α_{r}^{(j)} of F_{f} corresponds a root (α_{1}^{(j)}, …, α_{r}^{(j)}) of f̄; there are D solutions altogether all being ’finite’^{15} since R_{r+1} ≠ 0. Similarly to each of the D roots z^{(j)} of F_{f}^{(u)} corresponds a root (β_{1}^{(j)}, …, β_{r}^{(j)}, z^{(j)}) of f̂^{(u)}; clearly, up to a reenumerating we have

(α_{1}^{(j)}, …, α_{r}^{(j)}) = (β_{1}^{(j)}, …, β_{r}^{(j)})

and since f̂_{u}(β_{1}^{(j)}, …, β_{r}^{(j)}, z^{(j)}) = 0 we have z^{(j)} = ∑_{i=1}^{r} U_{i}α_{i}^{(j)} so that

F_{f}^{(u)} = R′_{r+1} ∏_{i=1}^{D} (Z − ∑_{i=1}^{r} U_{i}α_{i}^{(j)}) .

In conclusion

Proposition 41.6.2 (Macaulay). The u-resultant F_{f}^{(u)} is a product of D factors which are linear in Z, U_{1}, …, U_{r} and the coefficients of U_{1}, …, U_{r} in each factor supply a solution of the system f.

Also the number of solution is either D = ∏_{i−1}^{r} d_{i} or infinite, the latter

being the case when F_{f} vanishes identically.  �

15 Id est affine points (α_{1}^{(j)}, …, α_{r}^{(j)}) ∈ K^{r} corresponding to the projective point (1, α_{1}^{(j)}, …, α_{r}^{(j)}) ∈ ℙ^{r}(K).

Remark 41.6.3 (Macaulay). (1) Denoting D the determinant for the generic forms F̂_{1}, …, F̂_{r}, F̂_{u} regarding Z_{1}, …, Z_{r}, Z_{0} as variables and A its extraneous factor, we have D = A Res(F̂_{1}, …, F̂_{r}, F̂_{u}) and, setting Z_{0} = 0, that (Corollary 41.3.9(7)) A depends only on the coefficients of the generic polynomials χ_{r+1}(F_{1}), …, χ_{r+1}(F_{r}). Hence A is independent of V_{1}, …, V_{d} and U_{1}, …, U_{r}.

(2) In case of non-homogeneous polynomials the preliminary generic change of coordinates does not affect the homogeneity variable; thus it is possible for R_{r+1} to vanish identically. The consequence is a diminution in the number of finite solutions for Z but not in the number of linear factors of F_{f}; such factors have the shape ∑_{i=1}^{r} U_{i}α_{i} and correspond to an infinite

solution^{16} in the ratio α_{1} : α_{2} : ⋯: α_{r}.  �

In the generalized setting of Remarks 41.5.6 and 41.5.7, Macaulay’s result can be read as follows:

Proposition 41.6.4 (Lazard). Given h (non-homogeneous) polynomials

f_{1}, …, f_{h} ∈ Q, deg(f_{i}) := d_{i}, h ≥ r, d_{1} ≥ d_{2} ≥ ⋯ ≥ d_{h}, d = 1 − r + ∑_{i=1}^{r} d_{i},

and setting

• f_{h+1} := U_{0} + ∑_{i−1}^{r} U_{i}Z_{i},

• M ∈ K[Z_{0}, U_{0}, U_{1}, …, U_{r}] the Macaulay’s matrix constructed, according to Remark 41.5.6 and 41.5.7, via f_{1}, …, f_{h}, f_{h+1},

• R the corresponding Macaulay’s resultant,

• G := R(1, U_{0}, U_{1}, …, U_{r}) ∈ K[U_{0}, U_{1}, …, U_{r}],

we have

(1) f_{1}, …, f_{h} have a finite number of common roots if and only if M has rank \binom{d+r}{r} id est G ≠ 0;

(2) If G ≠ 0, deg(G) is the number of common roots of f_{1}, …, f_{h} counting multiplicity and zeros at infinitiy;

(3) G is homogeneous and, in K[U_{0}, U_{1}, …, U_{n}], is a product of lineear polynomials;

(4) if α_{0}U_{0} + α_{1}U_{1} + ⋯ + α_{n}U_{n} is a linear factor of G, then • if α_{0} ≠ 0 then (\frac{α_{1}}{α_{0}} , ⋯, \frac{α_{n}}{α_{0}} ) ∈ k^{n} is a root of the f_{i}s;

• if α_{0} = 0, (α_{0}, α_{1}, ⋯, α_{n}) is a common zero at infinity.  �

16 Id est a projective point (0, α_{1}, α_{2}, ⋯, α_{r}).

41.7 Kronecker’s Resolvent

Let us consider a finite set

F_{n} := {f_{1}^{(n)}, …, f_{s_{n}}^{(n)}} ⊂ P = k[X_{1}, …, X_{n}] = k[X_{1}, …, X_{n−1}][X_{n}]

of non-homogeneous polynomials generating an ideal I := 𝕀(F_{n}) which we assume to be in sufficiently ’generic’ position, the variables having been subjected to a change of coordinate beforehand. As a consequence, in particular, each f_{i}^{(n)} is regular in X_{n}^{17}.

Also I is in allgemeine position (Definition 34.4.3) so that for each primary component 𝔮 of I, dim(𝔮) = d, we have I ∩ k[X_{1}, …, X_{d}] = (0) thus, the construction of Chapter 39, page 13, can be compacted and extended: we can introduce the fields K_{d} := k(X_{1}, …, X_{d}) and their algebraic closures K_{d} ⊂ Ω(k) knowing that to each primary component 𝔮 of rank r = n − d the corresponding roots have the shape (X_{1}, …, X_{d}, β_{1}, …, β_{r}), β_{i} ∈ K_{d}.

We can iteratively, for ν := n, n − 1, .., 1, compute^{18}:

• D_{ν} := gcd(F_{ν}) ∈ k[X_{1}, …, X_{ν−1}][X_{ν}];

• g_{i}^{(ν)} := f_{i}^{(ν)}/D_{ν} ∈ k[X_{1}, …, X_{ν−1}][X_{ν}], 1 ≤ i ≤ s_{ν};

• G_{ν} := {g_{i}^{(ν)}, 1 ≤ i ≤ s_{ν}} ⊂ k[X_{1}, …, X_{ν−1}][X_{ν}];

• f := ∑_{i=1}^{s_{ν}} U_{i}g_{i}^{(ν)} ∈ k[X_{1}, …, X_{ν−1}][U_{1}, W_{1}, …, U_{s_{ν}} , W_{s_{ν}}][X_{ν}];

• g := ∑_{i=1}^{s_{ν}} W_{i}g_{i}^{(ν)} ∈ k[X_{1}, …, X_{ν−1}][U_{1}, W_{1}, …, U_{s_{ν}} , W_{s_{ν}}][X_{ν}];

• R_{ν} := Res(f, g) =: ∑_{υ∈U^{(sν)}} f_{υ}υ ∈ k[X_{1}, …, X_{ν−1}][U_{1}, W_{1}, …, U_{s_{ν}} , W_{s_{ν}}] where, for each value j ∈ ℕ, we use the notation

U^{(j)} := {U_{1}^{a_{1}} ⋯ U_{j}^{a_{j}}W_{1}^{b_{1}} ⋯ W_{j}^{b_{j}} : (a_{1}, …, a_{j}, b_{1}, …, b_{j}) ∈ ℕ^{2j}};

• F_{ν−1} := {f_{1}^{(ν−1)}, …, f_{s_{ν−1}}^{(ν−1)}} := {f_{υ}, υ ∈ U^{(s_{ν})}} ⊂ k[X_{1}, …, X_{ν−1}]

and remark that

(1) 1 = gcd(G_{ν}) ∈ k[X_{1}, …, X_{ν−1}][X_{ν}], so that

(2) R_{ν} ≠ 0;

(3) each f^{(ν)} is regular in X_{ν} since we are assuming that each variable has been subjected to a generic change of coordinate;

(4) each common root of F_{ν} is either a root of D_{ν} or a common root of G_{ν} and

(5) each common root α of G_{ν} is a common root of F_{ν−1} since R_{ν} ∈ 𝕀(f, g).

17 A polynomial f = ∑_{τ∈T c(f,τ} τ ∈ P, deg(f) = d is regular in X_{i} iff c(f, X_{i}^{d}) ≠ 0. 18 Compare Section 20.4

(6) On the other side if β := (X_{1}, …, X_{ν−1}, β_{1}), β_{1} ∈ K_{ν−1} is such that D_{ν}(X_{1}, …, X_{ν−1}, β_{1}) = 0 then R_{ν+1}(X_{1}, …, X_{ν−1}, β_{1}) = 0,

f := ∑_{i=1}^{s_{ν+1}} U_{i}g_{i}^{(ν+1)}(β_{1}, X_{ν+1}) and g := ∑_{i=1}^{s_{ν+1}} W_{i}g_{i}^{(ν+1)}(β_{1}, X_{ν+1})

have a common root β_{2} ∈ K_{ν−1} so that, for each i, 1 ≤ i ≤ s_{ν+1} we have

g_{i}^{(ν+1)}(X_{1}, …, X_{ν−1}, β_{1}, β_{2}) = 0 and f_{i}^{(ν+1)}(X_{1}, …, X_{ν−1}, β_{1}, β_{2}) = 0

(7) and, by iterating this argument, each root (X_{1}, …, X_{ν−1}, β_{1}) of D_{ν} lifts to a root (of rank n − ν + 1 and dimension ν − 1) of I.

Definition 41.7.1. The polynomial ∏_{1}^{n} D_{ν} is called the complete (total) resolvent of 𝕀(F_{n}); each factor D_{ν} is called the complete partial resolvent of

𝕀(F_{n}) of dimension ν − 1 and rank n − ν + 1.  �

Proposition 41.7.2. The complete resolvent of 𝕀(F_{n}) is a member of 𝕀(F_{n}).

Proof. In fact, for each ν there are

p_{ν}, q_{ν} ∈ k[X_{1}, …, X_{ν−1}][U_{1}, W_{1}, …, U_{s_{ν}} , W_{s_{ν}}][X_{ν}]

such that R_{ν} = p_{ν} (∑_{i=1}^{s_{ν}} U_{i}g_{i}^{(ν)}) + q_{ν} (∑_{i=1}^{s_{ν}} W_{i}g_{i}^{(ν)}) so that F_{ν−1} ⊂ 𝕀(G_{ν}) and

D_{ν}F_{ν−1} ⊂ 𝕀(F_{ν}) whence, by inductive argument F_{1} ∏_{ν=1}^{n} D_{ν} ⊂ 𝕀(F_{n}) where either F_{1} ∈ k or s_{1} > 1 and F_{1} = {f_{1}, …, f_{s_{1}}} ∈ k[X_{1}] with gcd(F_{1}) = 1 so that there are polynomials q_{i}(X_{1}) ∈ k[X_{1}] such that 1 = ∑_{i=1}^{s_{n}} q_{i}f_{i} and

∏_{ν=1}^{n} D_{ν} = ∑_{i=1}^{s_{n}} (q_{i} ∏_{ν=1}^{n} D_{ν}) f_{i} ∈ 𝕀(F_{n}).  �

Remark 41.7.3 (Macaulay). (1) As a direct consequence we have Hilbert’s

Nullstellensatz: if J has no root, the complete resolvent is 1 so that 1 ∈ I.

(2) Let us be given n forms f_{i} in n variables each of degree l which have no proper solution so that the complete resolvent is^{19} D_{1} = X_{1}^{μ}. Since the elements of F_{n} have all degree l, the elements of F_{n−1} have all degree δ^{2}, those of F_{n−2} degree (l^{2})^{2}; in general the terms of F_{ν} have all degree (l^{2^{ν−1}})^{2} = l^{2^{ν}} so that μ = l^{2^{n−1}}.

We should arrive at a similar result if we change x_{i} to x_{i} + a_{i} (i = 1, 2, …, n) beforehand, thus making the polynomials nonhomogeneous. The complete resolvent would be (x_{n} + a_{n})^{l^{2^{n−1}}} . The resultant would be (x_{n} + a_{n})^{l^{n}}. The difference in the two results is explained by the fact that the resultant is obtained by a process applying uniformly to all the variables, and the resolvent by a process applied to the variables in succession.^{20}

19 The construction reads the forms as polynomials and the sought roots are considered affine so if there is no proper solution, the origin is to be considered a root with a proper multiplicitly.

20 F. S. Macaulay, The Algebraic Theory op. cit., ppg. 21-2.

�

41.8 Kronecker: the u-resolvent

Given a basis F := {f_{1}, …, f_{s}} ⊂ k[X_{1}, …, X_{n}], let us consider new variables X, Λ_{1}, …, Λ_{n} such that X stands for

X = Λ_{1}X_{1} + ⋯ + Λ_{n}X_{n}

and perform the Liouville substitution^{21}

X_{i} = \{ \begin{matrix}\frac{X−Λ_{1}X_{1}+···+Λ_{n−1}X_{n−1}}{Λ_{n}} & if i = n \\ X_{i} & otherwise; \end{matrix}

thus obtaining

• the polynomials^{22}

f′_{i} := Λ_{n}^{l_{n}}F_{i} (X_{1}, … X_{n−1}, \frac{X − ∑_{1}^{n−1} Λ_{i}X_{i}}{Λ_{n}} ) , l_{n} := deg_{n}(F_{i}),

• the basis F′ := {f′_{1}, …, f′_{s}} ⊂ k[Λ_{1}, …, Λ_{n}][X_{1}, …, X_{n−1}][X],

• the ideal I′ := 𝕀(F′) ⊂ k[Λ_{1}, …, Λ_{n}][X_{1}, …, X_{n−1}, X].

Clearly there is a one-to-one correspondence between

• the roots (ξ_{1}, …, ξ_{n}) ∈ Z(F) and

• the roots (ξ_{1}, …, ξ_{n−1}, ξ) ∈ Z(F′)

the relation being given by ξ = Λ_{1}ξ_{1} + ⋯ + Λ_{n}ξ_{n}.

Definition 41.8.1. The complete resultant F_{u} := ∏_{1}^{n} D′_{ν} of 𝕀(F′) is called the complete u-resolvent of 𝕀(F).

21 Clearly

x := Λ_{1}x_{1} + ⋯ + Λ_{n}x_{n} ∈ k(Λ_{1}, …, Λ_{n})[x_{1}, …, x_{n}]
:= k(Λ_{1}, …, Λ_{n})[X_{1}, …, X_{n}]/𝕀(F)^{e}

is a primitive element in k[x_{1}, …, x_{n}] := k[X_{1}, …, X_{n}]/𝕀(F) for any ’generic’ evaluation of the Λ_{i}s.

22 The multiplier Λ_{n}^{l_{n}} being introduced to make [f′_{i}] integral in [Λ_{n}]. F. S. Macaulay, The Algebraic Theory op. cit., pg. 24. id est to grant that each f′_{i} ∈ k[Λ_{1}, …, Λ_{n}][X_{1}, …, X_{n}].

We have F_{u}(X_{1}, …, X_{n−1}, Λ_{1}X_{1} + ⋯ + Λ_{n}X_{n}) ∈ 𝕀(F) since (Proposition 41.7.2) F_{u} ∈ 𝕀(F′); moreover F_{u} considered as a univariate polynomial,

F_{u} ∈ k[Λ_{1}, …, Λ_{n}][X_{1}, …, X_{n−1}][X]

factors into linear factors, those of dimension ν − 1, id est the factors of the component D_{ν}, having the shape

X − Λ_{1}X_{1} − ⋯ − Λ_{ν−1}X_{ν−1} − Λ_{ν}ξ_{r} − ⋯ − Λ_{n}ξ_{1}.  (41.2)

Remark 41.8.2. The linear factors (41.2) of the complete partial resolvent D_{ν} is related to components of dimension d := ν − 1 and rank r := n − ν + 1.

According our notation the irreducible components

R ∈ k[Λ_{1}, …, Λ_{n}][X_{1}, …, X_{n−1}][X]

of D_{ν} should be read as elements R ∈ k[Λ_{1}, …, Λ_{n}][V_{1}, …, V_{d}, Z_{1}, …, Z_{r}][X] and the linear factors (41.2) as

X − Λ_{1}V_{1} − ⋯ − Λ_{ν−1}V_{d} − Λ_{d+1}ξ_{1} − ⋯ − Λ_{n}ξ_{r}.

According the notation introduced by Macaulay ^{23} and reported in Section 30.5 and here in Section 41.6, R must be read as an element

R ∈ k[Λ_{1}, …, Λ_{n}][x_{n}, …, x_{r+1}][x_{r}, …, x_{1}][X]

and the linear factors (41.2) as

X − Λ_{1}x_{n} − ⋯ − Λ_{d}x_{r+1} − Λ_{d+1}ξ_{r} − ⋯ − Λ_{n}ξ_{1}.

�

41.9 Kronecker Parametrization

In general the splitting factorization of D_{ν} could contain linear factors (41.2) where some ξ_{i} depends on the Λs.

Definition 41.9.1. A linear factor (41.2) of D_{ν} where each ξ_{i} is independent

of the Λs is called true^{24}.  �

23 To be more precise, the Liouville substitution performed by Macaulay was

x = u_{1}x_{1} + … + u_{n}x_{n}.

As we already pointed in 36.3, footnote 15, in order to adapt Macaulay’s notation to the current usage of chosing the first variables as parameters, one has to set x_{i} := X_{n−i}

24 In this case, in relation wiith Remark 41.8.2, the ξ_{i}s are elements, with our notations of the algebraic closure of K = k(V_{1}, …, V_{d}), with Macaualay’s of of the algebrtaic closure of k(x_{n}, …, x_{r+1}).

Remark 41.9.2 (Macaulay). Kronecker stated, without proving it, that each factor is true: “whether this is so or not must be considered doubtful.”^{25}

It could however be proved that

• a solution supplied by a non-true factor is necessarily embedded;

• any irreducible component R of a partial resultant D_{ν} either factors in true

linear factors only or has no true factor.  �

Historical Remark 41.9.3. Today the natural way for restricting ouselves to true factors is to get rid of embedded components via a radical computation; the more so since Macaulay already gave a procedure (Algorithm 30.7.3) for recovering embedded components and their multiplicity.

But I guess that Seidenberg’s Algorithm (Corollary 35.2.3) is the first

procedure proposed for radical computation.  �

So let us consider an irreducible component

R(X) ∈ k[Λ_{1}, …, Λ_{n}][X_{1}, …, X_{ν−1}][X]

of the partial resultant D_{ν} having a linear factorization into true factors of dimension d := ν − 1 and rank r := n − ν + 1:

R(X) = ∏_{j=1}^{δ} (X − Λ_{1}X_{1} − ⋯ − Λ_{ν−1}X_{ν−1} − Λ_{ν}ξ_{1j} − ⋯ − Λ_{n}ξ_{rj})
= (X − Λ_{1}V_{1} − ⋯ − Λ_{ν−1}V_{d} − Λ_{d+1}ξ_{1j} − ⋯ − Λ_{d+r}ξ_{rj})

where ξ_{ij} ∈ K_{ν−1}, and let us evaluate it at

Λ_{1}X_{1} + ⋯ + Λ_{n}X_{n} = Λ_{1}V_{1} + ⋯ + Λ_{d}V_{d} + Λ_{d+1}Z_{1} + Λ_{d+r}Z_{r}

obtaining

R′ := R(Λ_{1}V_{1} + ⋯ + Λ_{d}V_{d} + Λ_{d+1}Z_{1} + Λ_{d+r}Z_{r})
= ∏_{j=1}^{δ} (Λ_{d+1}(Z_{1} − ξ_{1j}) + ⋯ + Λ_{d+i}(Z_{i} − ξ_{ij}) + ⋯ + Λ_{n}(Z_{r} − ξ_{rj}))

so that R′ ∈ k[Λ_{ν}, …, Λ_{n}, Z_{1}, …, Z_{r}] = k[Z_{1}, …, Z_{r}][Λ_{ν}, …, Λ_{n}].

To [R] corresponds what is called an irreducible spread, viz. the spread of all points [ξ_{n}, …, ξ_{r+1}, ξ_{rj}, …, ξ_{1j}] in which [ξ_{n}, …, ξ_{r+1}] take all finite values, and [xi_{rj}, …, ξ_{1j}] the [δ] sets of values supplied by the linear factors of [R] which vary as [ξ_{n}, …, ξ_{r+1}] vary. […] No linear factor of [R] can be repeated, unless [X_{1}, …, X_{ν−1}] are given special values; for otherwise [R] and [\frac{∂R}{∂X} ] would have an H.C.F. involving [X], and [R] would be the product of two factors^{26}.

25 F. S. Macaulay, The Algebraic Theory op. cit., pg. 26. 26 F. S. Macaulay, The Algebraic Theory op. cit., pg. 27.

In other words, to the irreducible component R of the partial resultant D_{ν} corresponds a prime component 𝔣 := 𝔣_{R} of 𝕀(F) and an associated variety Z(R) := Z(𝔣_{R}) of dimension d := ν − 1 and rank r := n − d.

Moreover, in the expansion of R′ ∈ k[Z_{1}, …, Z_{r}][Λ_{ν}, …, Λ_{n}] the coefficient of each term in {Λ_{ν}^{a_{ν}} ⋯ Λ_{n}^{a_{n}} : (a_{ν}, …, a_{n}) ∈ ℕ^{r}}

all vanish at every point of the spread [Z(R)] and do not all vanish at any other point^{27}.

In particular the coefficient of

• Λ_{d+1}^{δ} is q(V_{1}, …, V_{d}, Z_{1}) := ∏_{j=1}^{δ}(Z_{1} − ξ_{1j}), ∈ k[V_{1}, …, V_{d}][Z_{1}]

• Λ_{d+i}Λ_{d+1}^{δ−1}, 1 < i ≤ r, is

q(V_{1}, …, V_{d}, Z_{1}) ∑_{j} \frac{Z_{i} − ξ_{ij}}{Z_{1} − ξ_{1j}}

= \frac{∂q}{∂Z_{1}} (V_{1}, …, V_{d}, Z_{1})Z_{i} − w_{i}(V_{1}, …, V_{d}, Z_{1}),

where w_{i}(V_{1}, …, V_{d}, Z_{1}) = q(V_{1}, …, V_{d}, Z_{1}) ∑_{j} \frac{ξ_{ij}}{Z_{1}−ξ_{1j}};

moreover we also have

q(V_{1}, …, V_{d}, Z_{1}) = \frac{∂q}{∂Z_{1}} (V_{1}, …, V_{d}, Z_{1})Z_{1} − w_{1}(V_{1}, …, V_{d}, Z_{1}),
w_{1}(V_{1}, …, V_{d}, Z_{1}) = q(V_{1}, …, V_{d}, Z_{1}) ∑_{j} \frac{ξ_{1j}}{Z_{1} − ξ_{1j}} .

Thus the roots (ξ_{1}, …, ξ_{r}) ∈ Z(R) satisfy the parametrization^{28}

\{ q(V_{1}, …, V_{d}, T) = 0, ; Z_{1} = \frac{w_{1}(V_{1},...,V_{d},T)}{\frac{∂q}{∂T} (V_{1},...,V_{d},T)} ; ⋮ ; Z_{r} = \frac{w_{r}(V_{1},...,V_{d},T)}{\frac{∂q}{∂T} (V_{1},...,V_{d},T)}  (41.3)

Definition 41.9.4. A parametrization (41.3) of a prime ideal

I ⊂ P, dim(I) = ν − 1,
in ’generic’ position is called a Kronecker parametrization of I.  �

27 F. S. Macaulay, The Algebraic Theory op. cit., pg. 27. 28 Where we have simply substituted Z_{1} with T.

41.10 *Historical Intermezzo: from Bézout to Cayley

In connection to Sylvester’s resultant, both Sylvester and Cayley quote^{29} Bézout’s abridged method to obtain the resultant or^{30} Bézout’s abbreviated method of elimination. Without pretending to give a survey on Bézout’s result, I think helpful to give some pointers to it for the interested reader.

His method for computing resultants is preliminarly described by Bézout^{31} in the case of three homogeneous linear equations^{32} ∑_{j=1}^{4} a_{ij}X_{j}, 1 ≤ i ≤ 3 : he considers the product ∏_{j=1}^{4} X_{j} and successively, for i = 1..3, substutes each X_{j} with a_{ij} observing the signe rule.^{33} We thus obtain

a11X2X3X4 − a12X1X3X4 + a13X1X2X4 − a14X1X2X3  i = 1
(a11a22 − a21a12)X3X4 − (a11a23 − a21a13)X2X4
+ (a11a24 − a21a14)X2X3 + (a12a23 − a22a13)X1X4 i = 2
−  (a12a24 − a22a14)X1X3 + (a13a24 − a23a14)X1X2
[(a11a22 − a21a12)a33 − (a11a23 − a21a13)a32 + (a12a23 − a22a13)a31] X4
−  [(a11a22 − a21a12)a34 − (a11a24 − a21a14)a32 + (a12a24 − a22a14)a31] X3
+ [(a11a23 − a21a13)a34 − (a11a24 − a21a14)a33 + (a13a24 − a23a14)a31] X2 i = 3
−  [(a12a23 − a22a13)a34 − (a12a24 − a22a14)a33 + (a13a24 − a23a14)a32] X1

whence he deduces

\{ \begin{matrix}X_{1} & = & X_{4} \frac{−[(a_{12}a_{23}−a_{22}a_{13})a_{34}−(a_{12}a_{24}−a_{22}a_{14})a_{33}+(a_{13}a_{24}−a_{23}a_{14})a_{32}]}{(a_{11}a_{22}−a_{21}a_{12})a_{33}−(a_{11}a_{23}−a_{21}a_{13})a_{32}+(a_{12}a_{23}−a_{22}a_{13})a_{31}} \\ X_{2} & = & X_{4} \frac{[(a_{11}a_{23}−a_{21}a_{13})a_{34}−(a_{11}a_{24}−a_{21}a_{14})a_{33}+(a_{13}a_{24}−a_{23}a_{14})a_{31}]}{(a_{11}a_{22}−a_{21}a_{12})a_{33}−(a_{11}a_{23}−a_{21}a_{13})a_{32}+(a_{12}a_{23}−a_{22}a_{13})a_{31}} \\ X_{3} & = & X_{4} \frac{−[(a_{11}a_{22}−a_{21}a_{12})a_{34}−(a_{11}a_{24}−a_{21}a_{14})a_{32}+(a_{12}a_{24}−a_{22}a_{14})a_{31}]}{(a_{11}a_{22}−a_{21}a_{12})a_{33}−(a_{11}a_{23}−a_{21}a_{13})a_{32}+(a_{12}a_{23}−a_{22}a_{13})a_{31}} \end{matrix}

id est Cramer’s formula.

As Muir^{34} put it

the unreal product ∏_{j=1}^{4} X_{j} at the very outset must have been a sore puzzle to students. […] To throw light upon the process, let us compare the above solution of a set of three linear equations with the following solution, which from one point of view may be looked upon as an improvement on the ordinary determinantal modes of solution as presented to modern readers.

29 J.J. Sylvester On a theory of the syzygietic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraic common measure. Phil . Trans. Royal Soc. London CXLIII (1853) pg. 407–548

30 A. Cayley, A fourth memory upon quantics Phil . Trans. Royal Soc. London

CXLVIII (1858) 415–427 31 E. Bézout Théorie generale des èquations algébriques (1771) Pierres, Paris, §200–

3 ppg.174–6. 32 I consider more suitable not to follow the original notation but properly adapt

it. 33 The reference being to Cramer’s rule of signes. 34 T. Muir The Theory of Detirminants in the Historical Order of Development

MacMillan (1906) London, pg. 44

[…] The numerators of the values of X_{1}, X_{2}, X_{3} and the common denominator are […] the coefficients of X_{1}, X_{2}, X_{3}, X_{4} in the determinant

| \begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ X_{1} & X_{2} & X_{3} & X_{4} \end{matrix} | := [X_{1}X_{2}X_{3}X_{4}]

More precisely, Muir explains, if we denote

[X_{i}X_{j}] := | \begin{matrix}a_{3i} & a_{3j} \\ X_{i} & X_{j} \end{matrix} | and [X_{i}X_{j}X_{h}] := | \begin{matrix}a_{2i} & a_{2j} & a_{2h} \\ a_{3i} & a_{3j} & a_{3h} \\ X_{i} & X_{j} & X_{h} \end{matrix} |

we have (by developing along the first line)

a_{11} [X_{2}X_{3}X_{4}] − a_{12} [X_{1}X_{3}X_{4}] + a_{13} [X_{1}X_{2}X_{4}] − a_{14} [X_{1}X_{2}X_{3}]

and, developing, again along the first line, the four determinants [X_{i}X_{j}X_{h}]

(a_{11}a_{22} − a_{21}a_{12}) [X_{3}X_{4}] − (a_{11}a_{23} − a_{21}a_{13})) [X_{2}X_{4}]
+ (a_{11}a_{24} − a_{21}a_{14})) [X_{2}X_{3}] + (a_{12}a_{23} − a_{22}a_{13})) [X_{1}X_{4}]
− (a_{12}a_{24} − a_{22}a_{14}) [X_{1}X_{3}] + (a_{13}a_{24} − a_{23}a_{14}) [X_{1}X_{3}]

and, finally, expanding the six determainants and recollecting the result

| \begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} | X_{4} − | \begin{matrix}a_{11} & a_{12} & a_{14} \\ a_{21} & a_{22} & a_{24} \\ a_{31} & a_{32} & a_{34} \end{matrix} | X_{3}

+ | \begin{matrix}a_{11} & a_{13} & a_{14} \\ a_{21} & a_{23} & a_{24} \\ a_{31} & a_{33} & a_{34} \end{matrix} | X_{2} − | \begin{matrix}a_{12} & a_{13} & a_{14} \\ a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \end{matrix} | X_{1}.

The same method, id est an expansion of proper determinants expressed with a similar notation and process, is then applied by Bézout to resolve different systems of polynomial equations, including^{35} computing the resultant in k[X_{1}] of two polynomials in k[X_{1}, X_{2}] and then is specialized to the computation of the resultant in k of two polynomials

35 Among the instances discussed, we can list the resultant

• in k[X] of a quadratic and a linear polynomial in k[X, Y] (§278–280, ppg. 215– 229);

• in k[X] of two polynomials XY − aX − bY − c (§281–284, ppg. 230–5);

• in k[X] of two quadratic polynomials in k[X, Y] (§285–91, ppg. 235–43; 303–5, ppg. 252–5);

• in k[X] of a quadratic and two linear polynomials in k[X, Y, Z] (§292, ppg. 244–5);

• in k[X] of three quadratic polynomials

aX^{2} + bXY + cXZ + dX + eY + fZ + g

in k[X, Y, Z] (§320, ppg. 269–71);

• in k of three polynomials XY − aX − bY − c (§373–4, ppg. 235–6);

ϕ := ∑_{i=0}^{n} a_{i+1}X^{n−i}, ϕ′ := ∑_{i=0}^{m} a′_{i+1}X^{m−i} ∈ k[X].

The specialized method, which is the one the English School called the Bézout’s abridged/abbreviated method, consists in

(1) multiplying ϕ and ϕ′ respectively by the polynomials

Φ := ∑_{i=0}^{ν} A_{i+1}X^{ν−i} and Φ′ := ∑_{i=0}^{μ} A′_{i+1}X^{μ−i}

whose degree (actually we have μ := n − 1 and ν := m − 1) is deduced by means of results similar to Theorem 41.2.3^{36};

(2) summing such two product, considering the linear system whose equations are the coefficients of the resulting polynomial and whose unknowns are the A_{i}, A′_{i}s and

(3) solving it by the method discussed above, id est via determinant expansion.

Example 41.10.1. Let us illustrate the easiest case of two quadratic univariate polynomials

ax^{2} + bx + c, a′x^{2} + b′x + c′ ∈ k[x] :

multiplying them by (respectively) Ax + B and A′x + B′ we obtain^{37}

une équation de cette forme

Aax^{3} + (Ab + Ba)x^{2} + (Ac + Bb)x + Bc = 0.

Egalant à zero le coefficient total de x^{3}, celui de x^{2}, &c. je procède au calcul de AA′BB′, comme il suit: Première ligne aA′BB′ Seconde ligne (ab′)BB′ − aA′aB′ Troisiéme ligne (ab′)bB′ − (ac′)aB′

• in k of three quadratic polynomials in k[X, Y] (§375, ppg. 326–8);

• in k of three quadratic polynomials in k[X] (§462, ppg. 389–90);

• in k of three cubic polynomials in k[X] (§463-4, ppg. 390–2).

36 The point is to reach a degree in which, equating the opportune coefficients of the terms in the equation ϕΦ + ϕ′Φ′ = 0, one obtains at least as many equations as unknowns. If the difference is positive leaving some freedom alternatively one can either equate some unknown to 0 or, in order to preserve symmetry, add equations of the shape a_{i}A′_{j} − a′_{i}A_{j} = 0 for convenient i, j (for an illustration compare Example 41.10.2).

37 Remark that Bézout uses the shorthand (ab′) to denote the determinant

| \begin{matrix}a & b \\ a′ & b′ \end{matrix} |.

en rejettant le terme où resteroit A′ qui n’étant point dans la dernière équation, ne peut plus influer sur l’equation finale. Quatriéme ligne (ab′)(bc′) − (ac′)^{2}. On a donc pour équation finale (ab′)(bc′) − (ac′)^{2} = 0.^{38}

If we expand Bézout’s computation using the same notation as above we have

aA′BB′ − Aa′BB′, (ab′ − a′b)BB′ − aA′aB′ + aA′Ba′ + Aa′aB′ − Aa′Ba′, (ab′ − a′b)(bB′ − Bb′) − (ac′ − ca′)(aB′ − Ba′) + (aA′ − Aa′)(ab′ − ba′), (ab′ − a′b)(bc′ − cb′) − (ac′ − ca′)^{2},

which can be interpretated as the expansion of the determinant

| \begin{matrix}a & a′ & 0 & 0 \\ b & b′ & a & a′ \\ c & c′ & b & b′ \\ 0 & 0 & c & c′ \\ A & A′ & B & B′ \end{matrix} |

corresponding to the linear system

\{ \begin{matrix}Aa + A′a′ & = & 0 \\ Ab + Ba + A′b′ + B′a′ & = & 0 \\ Ac + Bb + A′c′ + B′b′ & = & 0 \\ Bc + B′c′ & = & 0. \end{matrix}

Remark that in the last expansion the terms A and A′ are substituted by the corresponding coefficient 0 annihilating the last summand of the third expansion, justifying Bézout’s comment that the terms containing A′ (and A) can be removed, since in the fourth equations they don’t appear thus not influiencing the expansion.

Finally remark that the matrix whose determinant has been computed is

equivalent to Sylvester’s matrix | \begin{matrix}a & b & c & 0 \\ 0 & a & b & c \\ a′ & b′ & c′ & 0 \\ 0 & a′ & b′ & c′ \end{matrix} | .  �

Example 41.10.2. Let us now consider the system^{39}

\{ \begin{matrix}ax^{2} + bxy + cy^{2} + dx + ey + f & = & 0 \\ d′x + e′y + f′ & = & 0 \\ d”x + e”y + f” & = & 0 \end{matrix}

where the three equations are multiplied, respectively, by C, A′x + B′y + C′ and A”x + B”y + C”; considering, orderly, the coefficients of x^{2} and xy, the

38 E. Bézout op. cit., §347 pg.300. 39 E. Bézout op. cit., §369 ppg.319–20.

équation arbitraire B′d′ + B”d” = 0, the coefficients of y^{2}, x, y, 1 we have 7 equations in connection with 7 variables and we thus obtain the resultant

| \begin{matrix}d′ & d” & 0 & 0 & a & 0 & 0 \\ e′ & e” & d′ & d” & b & 0 & 0 \\ 0 & 0 & d′ & d” & 0 & 0 & 0 \\ 0 & 0 & e′ & e” & c & 0 & 0 \\ f′ & f” & 0 & 0 & d & d′ & d” \\ 0 & 0 & f′ & f” & e & e′ & e” \\ 0 & 0 & 0 & 0 & f & f′ & f” \end{matrix} |

which Bézout presents as

(d′e”) (c(d′f”)^{2} + (d′e”)(de′f”) − b(e′f”)(d′f”) + a(e′f”)^{2})

where he uses the shorthand notation (de′f”) = | \begin{matrix}d & e & f \\ d′ & e′ & f′ \\ d” & e′ & f” \end{matrix} |.  �

Historical Remark 41.10.3. A similar method is applied by Bézout also in order to compute resultants in k[X] of two polynomials in k[X, Y].

The main differences are that

(2) the linear system is obtained considering only the coefficients divisible by Y,

(3) the solution of the system returns the coefficients A_{i}, A′_{i} as rational functions in the a_{i}s and a′_{i}s;

(4) the resolvent is obtained by setting, in the polynomial obtained in step (2), Y = 0, id est removing the coefficients used in step (3), and substituting each A_{i}, A′_{i} with their expression in the a_{ij}s and a′_{i}s.

�

Example 41.10.4. Let us illustrate Bézout’s approach by considering^{40} the polynomials ax^{2} + bxy + cy^{2} + dx + ey + f and d′x + e′y + f′ which are respectively multiplied by F and D′x + E′y + F′ giving

(F a + D′d′)x^{2} + (F b + D′e′ + E′d)xy + (F c + E′e′)y^{2} (41.4)
+ (F d + D′f′ + F′d′)x + (F e + E′f + F′e′)y + fF

We then consider the equations (connected with y^{2}, xy and y)

\{ \begin{matrix}F c + E′e′ & = & 0 \\ F b + D′e′ + E′d & = & 0 \\ F e + E′f′ + F′e′ & = & 0. \end{matrix}

40 E. Bézout op. cit., §278 ppg.225–8.

and we expand the expression D′E′F F′ obtaining^{41}

−D′e′F F′ + D′E′cF′,
−e′e′F F′ + D′e′bF′ + e′E′cF′ − D′d′cF′,
−e′e′eF′ + e′e′F e′ − D′e′be′ + e′f′cF′ − e′E′ce′ + D′d′ce′,

and the solution (sic!) D′ = d′ce′ − e′be′, E′ = −e′ce′, F = e′e′e′, F′ = e′f′c; substituing it in (41.4) and setting y = 0 we obtain

(e′e′e′a + d′ce′d′)x^{2} + (e′e′e′d + d′ce′f′ + e′f′cd′)x + fe′e′e′
= e′ | \begin{matrix}c & bx + e & ax^{2} + dx + f \\ e′ & d′x + f′ & 0 \\ 0 & e′ & d′x + f′ \end{matrix} |

id est, up to the extraneous factor e′, the expected Sylvester resultant. �

The Sylvester resultant was, at least implicitly, introduced by Euler^{42}. His approach is essentially a variation of the one illustrated in Example 41.10.1: given

ϕ := ∑_{i=0}^{m} a_{i+1}(X)Y^{m−i}, ϕ′ := ∑_{i=0}^{n} a′_{i+1}(X)Y^{n−i} ∈ k(X)[Y]

he multiplies them, respectively, by

Φ := a′_{1}Y^{n−1} + ∑_{i=0}^{n−2} A_{i+1}Y^{n−2−i} and Φ′ := a_{1}Y^{m−1} + ∑_{i=0}^{m−2} A′_{i+1}Y^{m−2−i}

and subtracts the result, obtaining a polynomial of degree m + n − 2 — the coefficient of Y^{m+n−1} being 0; thus equating the coefficients of Y, Y^{2}, …, Y^{m+n−2} we obtain m + n − 2 linear equations into (m − 1) + (n − 1) variables; the solution is then substituted in the constant coefficient A_{n−1}a_{m+1} − A′_{m−1}a′_{n+1} giving E(X) ∈ k(X).

The equation system can be expressed as

M · (a′_{1}, A_{1}, …, A_{n−1}, a_{1}, A′_{1}, …, A′_{m−1})^{T} = (0, …, 0, E(X))^{T}

where M is the Sylvester matrix.

It is clear that the procedure proposed by Euler is equivalent to the computation of the Sylvester resultant, so that E(X) = Res(ϕ, ϕ′) is the required resultant.

41 Compare the corresponding matrix | \begin{matrix}0 & e′ & c & 0 \\ e′ & d′ & b & 0 \\ 0 & f′ & e & e′ \\ D′ & E′ & F & F′ \end{matrix} | .

42 L. Euler, Introductio in Analysin Infinitorum Tom. 2 (1748) Lausanne Chapter XIX, §483-5

In order to present the English School view of Bézout’s abridged method, I refer to Salmon’s Higher Algebra^{43}.

Given two polynomials of the same degree

U := ∑_{i=0}^{m} a_{i+1}X^{m−i}, ∈ k[X], V := ∑_{i=0}^{n} a′_{i+1}X^{n−i}, m = n

and denoting

V_{ρ} := ∑_{i=0}^{ρ} a′_{i+1}X^{ρ−i}, U_{ρ} := ∑_{i=0}^{ρ} a_{i+1}X^{ρ−i}, for each ρ, 0 ≤ ρ < n,

we compute the n polynomials of degree bounded by m − 1

F_{ρ} := V_{ρ−1}U − U_{ρ−1}V := ∑_{σ=1}^{m} α_{ρσ}X^{m−σ}, 1 ≤ ρ ≤ n;

we thus obtain the square matrix (α_{ρσ}) whose determinant is the required resultant; in order to extend this construction of a square matrix when m > n, we need to have e := m − n more polynomials of degree bounded by m − 1; the choice is to take F_{ρ} := X^{n−ρ}V, n < ρ ≤ m so that the resultant^{44}

is, therefore, as it ought to be, of the n^{th} degree in the coefficients of [U], and of the m^{th} in those of [V].

Example 41.10.5. It is sufficient to apply this recipe to the case of Example 41.10.1 to realize the equivalence with Bézout’s result.

We have

F_{1} = a(a′x^{2} + b′x + c′) − a′(ax^{2} + bx + c)
= (a, b′)x + (a, c′)
F_{2} = (ax + b)(a′x^{2} + b′x + c′) − (a′x + b′)(ax^{2} + bx + c)
= (a, c′)x + (b, c′)

whence the required determinant is | \begin{matrix}(ab′) & (ac′) \\ (ac′) & (bc′) \end{matrix} |.  �

43 G. Salmon, Lessons introductory to the Modern Higher Algebra, Fifth Ed.,

Chelsea Pub. Co. (1885) New York, §84-6, ppg. 81–3.

The method can be found in E. Bézout Recherches sur le degré des équations résultantes de l’évanouissement des inconnues, et sur les moyens qu’il convient d’employer pour trouver ses équations. Mém. Acad. Roy. Sci. Paris (1964) 288-338.

I was unable to read this paper so I rely to the description by Salmon and by H.K. Wimmer, On the History of the Bezoutian and the Resultant Matrix Linear Algebra and its Application 128 (1990) 27–34

44 Salmon, op. cit., §86, pg.83

The first which studied the bezoutic matrix B := (α_{ρσ}) was Jacobi^{45} which, among other comments, remarked that

• ∑_{ρσ} X^{ρ}α_{ρσ}X^{σ} = U(X)\frac{∂V (X)}{∂X} − V (X)\frac{∂U(X)}{∂X}^{46} (pg.103);

• B is symmetric (pg.102) and

• has no factore superfluo (pg.104);

• if det(B) ≠ 0 the inverse of B is a Hankel matrix^{47} (pg. 104);

• if det(B) = 0 the common roots α of U and V are in relations with the linear solutions (1, α, …, α^{n−1}) of B (pg. 104).

Sylvester^{48} gave in 1842 a formula^{49} to express the coefficients of the F_{ρ} (in case n = m) in terms of the determinants

(i, j) := (a_{i}a′_{j}) := | \begin{matrix}a_{i} & a_{j} \\ a′_{i} & a′_{j} \end{matrix} | , 1 ≤ i < j ≤ n + 1,

as follows:

[Conceive] a number of cubic blocks each of which has two numbers, termed its characteristics, inscribed upon one of its faces, upon which the values of such a block (itself called an element) depends. For instance, the value of the element, whose characteristics are r, s, is the difference between two products: the one of the coefficient rth

45 Jacobi, C.G.I., De eliminatione variabilis e duabus aequationibus algebraicas J.

Reine und Ang. Math. XV (1836) 101–24. 46 With the present notation, Jacobi remarked that

∑_{ρ=1}^{m} ∑_{σ=1}^{m} X^{m−ρ}α_{ρσ}X^{m−σ}

= ∑_{ρ=1}^{m} X^{m−ρ}F^{ρ}

= (∑_{ρ=1}^{m} X^{m−ρ} ∑_{i=0}^{ρ−1} a′_{i+1}X^{ρ−1−i}) U − (∑_{ρ=1}^{m} X^{m−ρ} ∑_{i=0}^{ρ−1} a_{i+1}X^{ρ−1−i}) V

= (∑_{i=0}^{m−1}(m − i)a′_{i+1}X^{m−i−1}) U − (∑_{i=0}^{m−1}(m − i)a_{i+1}X^{m−i−1}) V
= \frac{∂V (X)}{∂X} U − \frac{∂U(X)}{∂X} V.

47 id est the matrix det(B)^{−1}B^{−1} := (a_{ρσ}) satisfies, for each ρ, σ, a_{ρσ} = A_{ρ+σ−2} for

suitable A_{i}, 0 ≤ i ≤ 2n. 48 J.J. Sylvester Memoir on the dialytic method of elimination. Part I. Philosophical

Magazine XXXI (1842) 534–9 49 Sylvester’s matrix was given two years before: J.J. Sylvester A method of de-

termining by mere inspection the derivatives from two equations of any degree. Philosophical Magazine XVI (1840) 132–5

in order occurring in the polynomial U, by that which comes sth in order of the polynomial V; the other product is that of the coefficient sth in order of the polynomials U, by that rth in order of V; so that if the degree of each equation be n, there will be altogether \frac{1}{2}n(n+1) such elements. The blocks are formed into squares or flats (plafonds) of which the number is \frac{n}{2} or \frac{n+1}{2}, according as n is even or odd. The first of these contains n blanks in a side, the next (n−2), the next (n−4), till finally we reach a square of four blocks or of one, according as n is even or odd. These flats are laid upon one another so as to form a regularly ascending pyramid, of which the two diagonal planes are termed the planes of separation and symmetry respectively. The former divides the pyramid into two halves, such that no element on the one side of it is the same as that of any block in the other. The plan of symmetry, as the name denotes, divides the pyramid into two exactly similar parts; it being a rule, that all elements lying in any given line of a square (platfond) parallel to the plane of separation are identical; moreover the sum of the characteristics is the same for all elements lying anywhere in a plane parallel to that of separation.

The formula behind this rule is

α_{ρσ} = α_{σρ} = ∑_{j=σ+1}^{n+1}(a_{ρ+σ+1−j}, a′_{j}).  (41.5)

Example 41.10.6. For n = 2 we have the same formula as the one produced by Bézout (Example 41.10.1) (ab′ − a′b)(bc′ − cb′) − (ac′ − ca′)^{2}.

We illustrated Sylvester’s formula and construction for n = 3^{50}: we have

| 2, 3 | | \begin{matrix}1, 2 & 1, 3 & 1, 4 \\ 1, 3 & 1, 4 & 2, 4 \\ 1, 4 & 2, 4 & 3, 4 \end{matrix} |

giving the determinant

| \begin{matrix}(a_{1}a′_{2}) & (a_{1}a′_{3}) & (a_{1}a′_{4}) \\ (a_{1}a′_{3}) & (a_{1}a′_{4}) + (a_{2}a′_{3}) & (a_{2}a′_{4}) \\ (a_{1}a′_{4}) & (a_{2}a′_{4}) & (a_{3}a′_{4}) \end{matrix} |

50 The cases n = 4, 5 can be found in Salmon op. cit. §84-5, ppg. 81-2; all cases up

to 6 in J.J. Sylvester Memoir op. cit.

= | \begin{matrix}| \begin{matrix}a_{1} & a_{2} \\ a′_{1} & a′_{2} \end{matrix} | & | \begin{matrix}a_{1} & a_{3} \\ a′_{1} & a′_{3} \end{matrix} | & | \begin{matrix}a_{1} & a_{4} \\ a′_{1} & a′_{4} \end{matrix} | \\ | \begin{matrix}a_{1} & a_{3} \\ a′_{1} & a′_{3} \end{matrix} | & | \begin{matrix}a_{1} & a_{4} \\ a′_{1} & a′_{4} \end{matrix} | + | \begin{matrix}a_{2} & a_{3} \\ a′_{2} & a′_{3} \end{matrix} | & | \begin{matrix}a_{2} & a_{4} \\ a′_{2} & a′_{4} \end{matrix} | \\ | \begin{matrix}a_{1} & a_{4} \\ a′_{1} & a′_{4} \end{matrix} | & | \begin{matrix}a_{2} & a_{4} \\ a′_{2} & a′_{4} \end{matrix} | & | \begin{matrix}a_{3} & a_{4} \\ a′_{3} & a′_{4} \end{matrix} | \end{matrix} |

which is to be compared with Sylvester determinant

| \begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} & 0 & 0 \\ 0 & a_{1} & a_{2} & a_{3} & a_{4} & 0 \\ 0 & 0 & a_{1} & a_{2} & a_{3} & a_{4} \\ a′_{1} & a′_{2} & a′_{3} & a′_{4} & 0 & 0 \\ 0 & a′_{1} & a′_{2} & a′_{3} & a′_{4} & 0 \\ 0 & 0 & a′_{1} & a′_{2} & a′_{3} & a′_{4} \end{matrix} | .

�

If m > n the same formula is applied simply by expressing V as

V := ∑_{i=0}^{n} a′_{i+1}X^{n−i} = ∑_{i=0}^{m} b_{i+1}X^{m−i}

with b_{i+1} = \{ \begin{matrix}0 & 0 ≤ i < m − n \\ a′_{i+1−m+n} & m − n ≤ i ≤ m. \end{matrix}

In 1857 Cayley^{51} gave in Crelle

la forme la plus simple sous laquelle on peut présenter cette méthode. Pour éliminer [x]^{52} entre deux équations du nième degré

∑_{i=0}^{n} a_{i+1}x^{n−i} = 0, ∑_{i=0}^{n} a′_{i+1}x^{n−i} = 0

on n’a qu’a former l’équation identique

\frac{∑_{i=0}^{n} a_{i+1}x^{n−i} ∑_{i=0}^{n} a′_{i+1}y^{m−i} − ∑_{i=0}^{n} a′_{i+1}x^{m−i} ∑_{i=0}^{n} a_{i+1}y^{n−i}}{x − y}

= (y^{n−1}, y^{n−2}, …, 1) ( \begin{matrix}a_{0,0} & a_{1,0} & … & a_{n−1,0} \\ a_{0,1} & a_{1,1} & … & a_{n−1,1} \\ ⋮ & ⋮ & ^{…} & ⋮ \\ a_{0,n−1} & a_{1,n−1} & … & a_{n−1,n−1} \end{matrix} ) ( x^{n−1} ; x^{n−2} ; ⋮ ; 1 )

51 A. Cayley, Note sur la méthode d’élimination de Bezout J. Reine und Ang. Math.

LIII (1857) 366–7.

The result is however earlier. It is explicitly reported in 1853 by Sylvester in On a theory op. cit. § 62.

52 Cayley gives the formula for two homoigeneous forms; I adapt his formulas to the non-homogeneous case and I use a modern notation instead of the one introduced by him.

oú l’expression qui forme le second membre représente la fonction suivant

(a_{0,0}x^{n−1} + a_{1,0}x^{n−2} + ⋯ + a_{n−1,0}) y^{n−1}
+ (a_{0,1}x^{n−1} + a_{1,1}x^{n−2} + ⋯ + a_{n−1,1}) y^{n−2}
⋯
+ (a_{0,n−1}x^{n−1} + a_{1,n−1}x^{n−2} + ⋯ + a_{n−1,n−1});

le résultat de l’élimination sera

| \begin{matrix}a_{0,0} & a_{1,0} & … & a_{n−1,0} \\ a_{0,1} & a_{1,1} & … & a_{n−1,1} \\ ⋮ & ⋮ & ... & ⋮ \\ a_{0,n−1} & a_{1,n−1} & … & a_{n−1,n−1} \end{matrix} | .

Example 41.10.7. For n = 2 we have

\frac{(ax^{2} + bx + c)(a′y^{2} + b′y + c′) − (a′x^{2} + b′x + c′)(ay^{2} + by + c)}{x − y}

= (ab′ − ba′)xy + (ac′ − ca′)x + (ac′ − ca′)y + (bc′ − cb′)

again returning the matrix ( \begin{matrix}(ab′) & (ac′) \\ (ac′) & (bc′) \end{matrix} ) .

For n = 3 and U(x) = (ax^{3} + bx^{2} + cx + d), V (x) = (a′x^{3} + b′x^{2} + c′x + d′) we have

\frac{U(x)V (y) − V (x)U(y)}{x − y}

= (ab′)x^{2}y^{2} + (ac′)(x^{2}y + xy^{2}) + (ad′)x^{2}
+ ((ad′) + (bc′)) xy + (ad′)y^{2} + (bd′)x + (bd′)y + (cd′)

giving the determinant | \begin{matrix}(a, b′) & (ac′) & (ad′) \\ (ac′) & (ad′) + (bc′) & (bd′) \\ (ad′) & (bd′) & (cd′) \end{matrix} |.  �

Sylvester^{53}, in the case n = m, denotes the polynomials F_{ρ}, 1 ≤ ρ ≤ n the bezoutians of U and V and remarks that

The determinant formed by arranging in a square the n sets of coefficients of the n Bezoutians, and which I shall term the Bezoutian

53 J.J. Sylvester On a theory op. cit. § 5.

matrix^{54}, gives, as is well known, the Resultant (meaning thereby the Result in its simplest form of eliminating the variables out) of U and V. Eliminating dialytically, first X^{n−1} between the first and the second, then X^{n−1} and X^{n−2} between the first, second and the third, and so on, and finally, all the powers of X between the first, second, third,…,nth of these Bezoutians, and repeating the first of them, we obtain a derived set of n equations, the right-hand members of which I shall term the secondary Bezoutains to U and V.

The ’dialytical elimination’ performed by Sylvester on the expressions

V_{ρ−1}U − U_{ρ−1}V = F_{ρ}, 1 ≤ ρ ≤ m

returns

V_{0}U − U_{0}V  = F_{1}  =: B_{1}
(α_{21}V_{0} − α_{11}V_{1})U − (α_{21}U_{0} − α_{11}U_{1})V = α_{21}F_{1} − α_{11}F_{2} =: B_{2}

…

S_{ρ}U − T_{ρ}V  =: B_{ρ}

…

S_{m−1}U − T_{m−1}V  =: B_{m−1}.

where we have deg(S_{ρ}) = deg(T_{ρ}) = ρ − 1, deg(B_{ρ}) = m − ρ; thus, assuming U, V to be^{55}

perfectly unrelated, and each the most general function that can be formed of the same degree

and in case m = n^{56} then if we repeatedly perfom the Division Algorithm and change the sign of each remainder, as in Section 13.3, we obtain the

54 But he also uses the term Bezoutic square. The term than stabilizies as bezoutic matrix (Compare Cayley’s entry Mathematics, recent terminology in in the English Cyclopædia, vol. V (1860) pgg.534-42). In particular Cayley (A fourth memory op.cit. § 88) labels the Bezoutic Emanant of U and V the polynomial

V_{ρ} := ∑_{i=0}^{ρ} a′_{i+1}X^{ρ−i}, U_{ρ+e} := ∑_{i=0}^{ρ+e} a_{i+1}X^{ρ+e−i}, for each ρ, 0 ≤ ρ < n

\frac{U(x)V (y)−V (x)U(y)}{x−y} introduced by him.

The term bezoutiant in fact, as we will see below, has been already associated to the quadratic function which (A. Cayley, A fourth memory op.cit. § 91)

Professor Sylvester forms with the matrix of the Bezoutic emanant.

55 J.J. Sylvester On a theory op. cit. §1 56 This argument and construction is in § 5. In the sequent § 6, Sylvester explains

how to extend it in the case m = n + e, e > 0.

He defines and computes the n polynomials (all of degree bounded by m − 1)

polynomial sequence^{57} U, V, R_{2}, …, R_{m} where each R_{i} necessarily satisfies deg(R_{i}) = m − i + 1. Thus, the^{58}

n successive Secondary Bezoutians to the system U, V […] will (saving at least a numerical factor of a magnitude and algebraic sign to be determined, but which, when proper conventions are made, will be subsequently proved to be +1) represent the simplified […] residue[s] to \frac{U}{V}.

id est B_{ρ} = R_{ρ+1} for each ρ.

Once obtained the Bezoutic square of two polynomials f, ϕ of the same degree m, Sylvester remarks that^{59}

this square […] is symmetrical about one of its diagonals, and corresponds therefore (as every symmetrical matrix must do) to a homogeneous quadratic function of m variables of which it expresses the determinant. This quadratic function, which plays a great part in […] the theory of real roots, I term the Bezoutiant. […] In Section V. Arts. 56.57, I show that the total number of effective intercalations between the roots of two functions of the same degree is given by the inertia of that quadratic form^{60} which we agreed to

F_{ρ} := V_{ρ−1}U − U_{ρ−1}V := ∑_{σ=1}^{m} α_{ρσ}X^{m−σ}, 1 ≤ ρ ≤ n;

next he introduces the e polynomials X^{μ}V, 0 ≤ μ < e and, for ρ, 1 ≤ ρ ≤ n, using these e polynomials and the ρ polynomials F_{r}, 1 ≤ r ≤ ρ he produces a relation

S_{ρ}U − T_{ρ}V = B_{ρ}, deg(S_{ρ}) = ρ − 1, deg(T_{ρ}) = e + ρ − 1, deg(B_{ρ}) = m − ρ;

thus the argument given in case n = m applies verbatim. 57 which is the Sturm sequence (Definition 13.3.1) if V = U′. 58 J.J. Sylvester On a theory op. cit. §5 59 J.J. Sylvester On a theory op. cit. Introduction 60 Sylvester introduced the notion of inertia and proved its Law of Inertia in On a

theory op. cit. § 44-5.

Recall that given a quadratic form f(x_{1}, …, x_{n}) = ∑_{i} ∑_{j} β_{ij}x_{i}x_{j}, β_{ij} = β_{ji} and denoting, for each two vectors u = (c_{1}, …, c_{n}), v = (d_{1}, …, d_{n}) in k^{n}

f(u, v) := ∑_{i} ∑_{j} β_{ij}c_{i}d_{j}

then the vectorspace N := {w ∈ k^{n} : f(w, u) = 0 for each u ∈ k^{n}} is invariant for linear transformation and such is also its dimension n − r.

Thus k^{n} has an orthogonal basis v_{1}, …, v_{r}, v_{r+1}, …, v_{n} so that

N = Span_{k}(v_{r+1}, …, v_{n}) and f(v_{i}, v_{j}) = \{ \begin{matrix}0 & i ≠ j \\ γ_{i} ≠ 0 & i = j ≤ r \\ 0 & i = j > r \end{matrix}

term the Bezoutiant to f and ϕ; and in the following article (58) the result is extended to embrace the case contemplated in M.Sturm’s theorem; that is to say, I show, that on replacing the function of x by a homogeneous function of x and y, the Bezoutiant of the two functions, which are respectively the differential derivates of f with respect to x and with respect to y, will serve to determine by its form or inertia the total number of real roots and of equal roots in f(x)^{61}. The subject is pursued in the following Arts. 59,60. […] In Arts. 61, 62, 63, it is proved that the Bezoutiant is an invariative function of the functions from which it is derived; and in Art. 64 the important remark is added, that it is an invariant of that particular class to which I have given the name of Combinants, which have the property of remaining unaltereted, not only for linear transformations of the variables, but also for linear combinations of the functions containing the variables^{62}, possessing thus a character of double invariability. In Arts. 65, 66 I consider the relation of the Bezoutiant to the differential determinant, so called by Jacoby, but which for greater brevity I call

so that for each u = ∑_{i} c_{i}v_{i} ∈ k^{n} we have f(u, u) := ∑_{i} c_{i}^{2}γ_{i}.

Sylvester’s Law of Inertia states that, if k = ℝ, the ’number of integers in the excess of positive over negative signs which adheres to a quadratic form expressed as the sum of positive and negative squares’ (which Sylvester names the inertia of the quadratic form) is ’unchangeable notwithstanding any real linear transformation impressed upon such form’ id est the inertia is the invariant

#{γ_{i} > 0, 1 ≤ i ≤ r} − #{γ_{i} < 0, 1 ≤ i ≤ r}.

61 In other words, Sylvester

• considers the polynomial f(x) = ∑_{i=0}^{m} a_{i}x^{m−i} and its derivate f′(x),

• performs Division Algorithm obtaining

f_{1}(x) = mf(x) − xf′(x) = ∑_{i=1}^{m} ia_{i}x^{m−i}

• computes the Bezoutian secondaries of f_{1} and f′, B_{1}, …, B_{m−1} which in this case are exactly the Sturm sequence and

• evaluates ’the number of pairs of imaginary roots in f(x)’ by counting ’the number of variations of sign betwen consecutive terms’ obtained evaluating f_{1}, f′, B_{1}, …, B_{m−1} at +∞.

Remark that, setting g(x, y) = ∑_{i=0}^{m} a_{i}x^{m−i}y^{i} we actually have

\frac{∂g}{∂y} = ∑_{i=1}^{m} ia_{i}x^{m−i}y^{i−1}, f_{1}(x) = \frac{∂g}{∂y} (x, 1)

justifying Sylvester’s reference to the derivate with respect to y. 62 Id est he considers the Bezoutiant of the functions kf + iϕ and k′f + i′ϕ and

remarks that each entry on the Bezoutic matrix is multiplied by (ki′ − k′i) so that the Bezoutiant (§64)

the Jacobian. On proper substitutions being made in the Bezoutiant for the m variables which it contains […], the Bezoutiant becomes identical with the Jacobian of f and Φ.

To illustrate the ‘proper substitution to be done’ I give again the word to Sylvester^{63}

[The Bezoutiant] B(u_{1}, …, u_{m}) being a covariant of the system f and ϕ […] on making u_{1}, …, u_{m} equal to [x^{m−1}, x^{m−2}y, …, y^{m−1}], B will become […] what I am in the habit of calling the Jacobian (after the name of the late but ever-illustrious Jacobi), a term capable of application to any number of homogeneous functions of as many variables. In the case before us, where we have two functions of two variables, the Jacobian

J(f, ϕ) = | \begin{matrix}\frac{df}{dx}, & \frac{dϕ}{dx} \\ \frac{df}{dy} , & \frac{dϕ}{dy} \end{matrix} | = \frac{df}{dx} \frac{dϕ}{dy} − \frac{df}{dy} \frac{dϕ}{dx}.

[…] So in the case of a single function F of the degree m, the Bezoutoid, that is the Bezoutiant to \frac{dF}{dx} , \frac{dF}{dy}, on making the (m − 1) variables which it contains identical with x^{m−2}, x^{m−3}y, …, y^{m−2} respectively, becomes identical with the Jacobian to \frac{dF}{dx} , \frac{dF}{dy}, that is the Hessian of F, namely

| \begin{matrix}\frac{d^{2}F}{dx^{2}} , & \frac{d^{2}F}{dxdy} \\ \frac{d^{2}F}{dxdy} , & \frac{d^{2}F}{dy^{2}} \end{matrix} | .

As an example of this property of the Bezoutiant, suppose

f = ax^{3} + bx^{2}y + cxy^{2} + dy^{3},
ϕ = αx^{3} + βx^{2}y + γxy^{2} + δy^{3}.

The Bezoutiant matrix becomes

\begin{matrix}aβ − bα, & aγ − cα, & aδ − dα, \\ aγ − cα, & ( \begin{matrix}aδ − dα \\ + \\ bγ − cβ \end{matrix} ) , & bγ − cβ, \\ aδ − dα, & bγ − cβ, & cδ − dγ. \end{matrix}

The Bezoutiant accordingly will be the quadratic function

becomes increased in the ratio of (ki′ − k′i)^{m}, that is remains always unalterated in point of form and absolutely immutable, provided that ki′ − k′i be taken, as we may always suppose to be the case, equal to 1.

63 J.J. Sylvester On a theory op. cit. § 65

(aβ − bα)u_{1}^{2} + {(aδ − dα) + (bγ − cβ)} u_{2}^{2} + (cδ − dγ)u_{3}^{2}
+ 2(aγ − cα)u_{1}u_{2} + 2(aδ − dα)u_{3}u_{1} + 2(bγ − cβ)u_{2}u_{3},

which on making

u_{1} = x^{2}, u_{2} = xy, u_{3} = y^{2}

becomes

Lx^{4} + Mx^{3}y + Nx^{2}y^{2} + P xy^{3} + Qy^{4},

where L, M, N, P, Q respectively will be the sum of the terms lying in the successive bands drawn parallel to the sinister diagonal of the Bezoutiant matrix, that is

L = (aβ − bα),
M = 2(aγ − cα),
N = 3(aδ − dα) + (bγ − cβ),
P = 2(bγ − cβ),
Q = (cδ − dγ).

The biquadratic function in x and y[…] will be found on computation to be identical in point of form with the Jacobian to f,ϕ, namely

(3ax^{2} + 2bxy + cy^{2})(βx^{2} + 2γxy + 3δy^{2}) − (3αx^{2} + 2βxy + γy^{2})(bx^{2} + 2cxy + 3dy^{2})

this latter being in fact

3Lx^{4} + 3Mx^{3}y + 3Nx^{2}y^{2} + 3P xy^{3} + 3Qy^{4}.

and concludes commenting:

The remark is not without some interest, that in fact the Bezoutiant, which is capable (as has been shown already) of being mechanically constructed, gives the best and readiest means of calculating the Jacobian; for in summing the sinister bands transverse to the axis of symmmetry the only numerical operation to be performed is that of addition of positive integers, whereas the direct method involves the necessity of numerical subtractions as well as additions, inasmuch as the same terms will be repeated with different signs.

and remarks, in a different example, that, unlike the computation via Bezoutiant, the direct evaluation requires to effectively employ also division in order to reduce the Jacobian to its simplest form, being divisible by deg(f) = deg(ϕ).

41.11 Dixon’s Resultant

The computation of a resultant of r forms in r variables was already solved by Bézout as an instance of this general approach.

An alternative proposal was put forward by Cayley^{64} based on what today we could call a solution via linear syzygies.

He assumes to have m_{1} variables connected by m_{2} linear equations, not being all independent, but connected by m_{3} linear equations, again not necessarily linearly independent; we thus obtain s matrices M_{σ} = (a_{ij}^{(σ)}), the

i^{th} matrix having m_{σ} columns and m_{σ+1} rows, the m_{σ}s being related by ∑_{σ=1}^{s+1}(−1)^{σ}m_{σ} = 0:

the number of quantities [m_{1}] will be equal to the number of really independent equations connecting them, and we may obtain by the elimination of these quantities a result Δ = 0.

The approach, denoting μ_{ϱ} := ∑_{σ=ϱ}^{s+1}(−1)^{σ−ϱ}m_{σ} = 0, consists in

• selecting μ_{s+1} = m_{s+1} indexes I_{s} ⊂ {1, …, m_{s}} and computing the determinant Q_{s} of the μ_{s+1}–square minor of M_{s} obtained by selecting the rows indexed by I_{s};

• selecting μ_{s} = m_{s} − μ_{s+1} indexes I_{s−1} ⊂ {1, …, m_{s−1}} and computing the determinant Q_{s−1} of the μ_{s}–square minor of M_{s−1} obtained by selecting the rows indexed by I_{s−1} and the columns indexed by {1, …, m_{s}} − I_{s}

• …

• selecting μ_{ϱ} = m_{ϱ} −μ_{ϱ+1} indexes I_{ϱ−1} ⊂ {1, …, m_{ϱ−1}} and computing the determinant Q_{ϱ−1} of the μ_{ϱ}–square minor of M_{ϱ−1} obtained by selecting the rows indexed by I_{ϱ−1} and the columns indexed by {1, …, m_{ϱ}} − I_{ϱ}

• …

• selecting μ_{3} = m_{3} − μ_{4} indexes I_{2} ⊂ {1, …, m_{2}} and computing the determinant Q_{2} of the μ_{2}–square minor of M_{2} obtained by selecting the rows indexed by I_{2} and the columns indexed by {1, …, m_{3}} − I_{3}

• computing, on the basis of the remark that m_{1} = μ_{2} = m_{2} − μ_{3}, the determinant Q_{1} of the μ_{2}–square minor of M_{1} obtained by selecting the columns indexed by {1, …, m_{2}} − I_{2};

finally, if each Q_{i} is not zero, one obtaind Δ by computing

Δ = Q_{1}Q_{2}^{−1}Q_{3}Q_{4}^{−1} ⋯ = ∏_{σ=1}^{s} Q_{σ}^{(−1)^{σ−1}}.

The application considers a set of forms {f_{1}, …, f_{u}} and, fixed a proper degree d ∈ ℕ intends to eliminate all terms of degree d among the equations F = 0 where F runs among the forms in the set

64 A. Cayley, On the theory of elimination Cambridge and Dublin Math. J. III (1848) 116-20

F := {τf_{i}, 1 ≤ i ≤ u, τ ∈ T , deg(τf_{i}) = d};

it consists in computing a linear resolution of the elements in F and applying on it the computation suggested above. Cayley however remarks that

I am not in possession of any method of arriving at once at the final result in its more simplified form; my process, on the contrary, leads me to a result encumbered by an extraneous factor, which is only got rid of by a number of successive divisions.

The first solution, apart Bèzout, for computing the resultant of more than 2 polynomials is due to A.L. Dixon^{65} which generalized Cayley’s interpretation of the Bezoutic/Bezoutian matrix in terms of the Bezoutic Emanant, proposing such Emanant for 3 polynomials in two variables and remarking that the constuction easily generalizes to polynomials in any number of variables.

Given three polynomials

ϕ(X_{1}, X_{2}) = ∑_{r=1}^{n} ∑_{s=1}^{m} A_{rs}X_{1}^{r}X_{2}^{s},

ψ(X_{1}, X_{2}) = ∑_{r=1}^{n} ∑_{s=1}^{m} B_{rs}X_{1}^{r}X_{2}^{s},

χ(X_{1}, X_{2}) = ∑_{r=1}^{n} ∑_{s=1}^{m} C_{rs}X_{1}^{r}X_{2}^{s}

Dixon considers the determinant

Δ := | \begin{matrix}ϕ(X_{1}, X_{2}) & ψ(X_{1}, X_{2}) & χ(X_{1}, X_{2}) \\ ϕ(X_{1}, Y_{2}) & ψ(X_{1}, Y_{2}) & χ(X_{1}, Y_{2}) \\ ϕ(Y_{1}, Y_{2}) & ψ(Y_{1}, Y_{2}) & χ(Y_{1}, Y_{2}) \end{matrix} |

and, remarking that it vanishes if we put X_{1} = Y_{1} and also if we put X_{2} = Y_{2}, and so it is divisible by (X_{1} − Y_{1})(X_{2} − Y_{2}), he considers the polynomial

D(X_{1}, X_{2}, Y_{1}, Y_{2}) = \frac{Δ(X_{1}, X_{2}, Y_{1}, Y_{2})}{(X_{1} − Y_{1})(X_{2} − Y_{2})}

which is of degree \{ \begin{matrix}2n − 1 & in X_{1} \\ m − 1 & in X_{2} \\ n − 1 & in Y_{1} \\ 2m − 1 & in Y_{2} \end{matrix} so that

Soc. 7 (1908) 49–69

65 The eliminant of three quatics in two independent variables, Proc. London Math.

equating to zero the cofficients of [Y_{1}^{r}Y_{2}^{s}], for all values of r and s, [D = 0] is equivalent to 2mn equations in [X_{1}, X_{2}] and the number of terms in these equations is also 2mn. Thus the eliminant^{66} can be at once written down as a determinant of order 2mn, each constituent of which is the sum of determinants of the third order of the type

Δ := | \begin{matrix}A_{pq} & A_{rs} & A_{tu} \\ B_{pq} & B_{rs} & B_{tu} \\ C_{pq} & C_{rs} & C_{tu} \end{matrix} |

In other words, denoting 𝔞 := {X_{1}^{r}X_{2}^{s}; r < 2n, s < m}, and 𝔟 := {Y_{1}^{r}Y_{2}^{s}; r < n, s < 2m}, we have

D(X_{1}, X_{2}, Y_{1}, Y_{2}) = ∑_{τ∈𝔞} ∑_{υ∈𝔟} d_{τυ}τυ, d_{τυ} ∈ ℤ[A_{pq}, B_{rs}, C_{tu}].

Clearly the vanishing of the determinant of the matrix (d_{τυ}) is equivalent of the existence of a common root of ϕ, ψ, χ.

Finally Dixon remarks that such method is

applicable to the problem of elimination when the number of variables is greater than two.

Denote, for each i, 0 ≤ i ≤ n,

g(X_{i}) := g(Y_{1}, …, Y_{i}, X_{i+1}, …, X_{n}) for each g(X_{1}, …, X_{n}) ∈ P,

so that, in particular g(X_{0}) = g(X_{1}, …, X_{n}) and g(X_{n}) = g(Y_{1}, …, Y_{n}).

Given n + 1 polynomials f_{1}, …, f_{n+1} ∈ k[X_{1}, …, X_{n}] each of degree n_{i} in the variable X_{i}, one can consider the determinant ^{67}

Δ := | \begin{matrix}f_{1}(X_{0}) & … & f_{n+1}(X_{0}) \\ f_{1}(X_{1}) & … & f_{n+1}(X_{1}) \\ ⋮ & ... & ⋮ \\ f_{1}(X_{i}) & … & f_{n+1}(X_{i}) \\ ⋮ & ... & ⋮ \\ f_{1}(X_{n}) & … & f_{n+1}(X_{n}) \end{matrix} |  (41.6)

which is divisible by ∏_{i=1}^{n}(X_{i} − Y_{i}) giving a polynomial

D(X_{1}, X_{2}, …, X_{n}, Y_{1}, …, Y_{n})

of degree m_{i} := (n + 1 − i)n_{i} − 1 in X_{i} and μ_{i} := in_{i} − 1 in Y_{i} so that

66 Salmon used the term eliminant to denote what we call resultant. 67 It is Dixon himself which reversed the order in which the variables are trans-

formed from X to Y; for two variables he transformed from right to left; in the final remark he makes the example of four polynomials in three variables and tranforms them from left to right.

D(X_{1}, X_{2}, …, X_{n}, Y_{1}, …, Y_{n}) = ∑_{τ∈𝔞} ∑_{υ∈𝔟} d_{τυ}τυ

where 𝔞 := {X_{1}^{a_{1}} … X_{n}^{a_{n}} : a_{i} ≤ m_{i}}, and 𝔟 := {Y_{1}^{α_{1}} … Y_{n}^{α_{n}} : α_{i} ≤ μ_{i}} and

#𝔞 = #𝔟 = n! ∏_{i=1}^{n} n_{i} := s.

Definition 41.11.1 (Kapur–Saxena–Yang). The polynomial D is called the Dixon polynomial of f_{1}, …, f_{n+1}.

The matrix D := (d_{τυ}) is called the Dixon matrix and its determinant the

Dixon resultant.  �

Remark 41.11.2 (Kapur–Saxena–Yang). Let D := (c_{τυ}) be the Dixon matrix of f_{1}, …, f_{n+1} and let us enumerate the elements of 𝔞 as

τ_{1} := 1, τ_{2} := X_{1}, …, τ_{n+1} := X_{n}, τ_{n+2}, …, τ_{s}.

If α := (a_{1}, …, a_{n}) ∈ Z(f_{1}, …, f_{n+1}) then

(1, a_{1}, …, a_{n}, τ_{n+2}(α), …, τ_{s}(α))
is a solution of the linear system D (τ_{1}, …, τ_{s})^{T}.  �

Remark 41.11.3. If n = 1 Dixon matrix and polynomial coincide with what Sylvester and Cayley called the bezoutic (or bezoutian) matrix and bezoutic

eminent.  �

Remark 41.11.4 (Kapur–Saxena–Yang). Dixon considered ’generic’ polynomials all having the same degree in each variable; as a consequence the Dixon matrix is square and one can speaks of determinant and introduce the Dixon resultant.

In specific instances, the default approach is the classical one already used by Sylvester and Cayley, namely assuming that the missing terms have coefficient zero.

For an alternative solution based on restricting oneself to complete inter-

section ideals, see below.  �

In a previous paper^{68} Dixon gave another interesting computational approach to evaluate the resultant^{69} in terms of Cayley’s formula: given two polynomials of the same degree^{70}

68 Dixon, A.L. On a form of the eliminant of two quatics, Proc. London Math. Soc.

6 (1908) 468–78 69 Which he called Bezout’s determinant. 70 A polynomial of lower degree is forced, as usual to reach the highest degree by

adding 0X^{m+1} + … + 0X^{n}.

U := ∑_{i=0}^{n} a_{i+1}X^{n−i}, ∈ k[X], V := ∑_{i=0}^{n} a′_{i+1}X^{n−i},

and denoting C(X, Y) := \frac{U(X)V (Y)−V (X)U(Y)}{X−Y} he states that Lemma 41.11.5. It holds

Res(U, V) = Δ := | \begin{matrix}d_{11} & ⋯ & d_{1n} \\ ⋮ & ... & ⋮ \\ d_{n1} & ⋯ & d_{nn} \end{matrix} |

where

d_{i,j} = \frac{1}{i!j!} \frac{∂^{i+j}C(X, Y)}{∂X^{i}∂Y^{j}} |_{X=Y =0} .

Proof. One has

C(X, Y) = ∑_{p>q}(a_{n−p+1}a′_{n−q+1} − a′_{n−p+1}a_{n−q+1})\frac{X^{p}Y^{q} − Y^{p}X^{q}}{X − Y}
= ∑_{p>q}(a_{n−p+1}a′_{n−q+1} − a′_{n−p+1}a_{n−q+1}) ( ∑_{i=0}^{p−q−1} X^{p−1−i}Y^{q+i});

thus d_{i,j}, which is the coefficient of X^{i}Y^{j} in C(X, Y) satisfies d_{i,j} = α_{ij} where α_{ij} is the result of Sylvester’s construction (41.5). �

He then fixes “two sets of arbitrary quantities” x_{1}, …, x_{n} and y_{1}, …, y_{n} and states Proposition 41.11.6 (Dixon). It holds

Res(U, V) = \frac{| \begin{matrix}C(x_{1}, y_{1}) & ⋯ & C(x_{1}, y_{n}) \\ ⋮ & ... & ⋮ \\ C(x_{n}, y_{1}) & ⋯ & C(x_{n}, y_{n}) \end{matrix} |}{∏_{i>l}(x_{i} − x_{j}) ∏_{i>l}(y_{i} − y_{j})}

Proof. If we expand each C(x_{i}, y_{i})

in ascending powers of (x_{i} − X) by Taylor’s theorem and the result in ascending powers of (y_{j} − Y) we have

| \begin{matrix}C(x_{1}, y_{1}) & ⋯ & C(x_{1}, yn) \\ ⋮ & ^{^{.}^{.} .} & ⋮ \\ C(xn, y_{1}) & ⋯ & C(xn, yn) \end{matrix} |
=  | \begin{matrix}1 & (y1 − Y ) & ⋯ & (y1 − Y )n−1 \\ ⋮ & ⋮ & ^{…} & ⋮ \\ 1 & (yn − Y ) & ⋯ & (yn − Y )n−1 \end{matrix} | · | \begin{matrix}1 & (x1 − X) & ⋯ & (x1 − X)n−1 \\ ⋮ & ⋮ & ^{…} & ⋮ \\ 1 & (xn − X) & ⋯ & (xn − X)n−1 \end{matrix} | · Δ
=  ∏_{i>l}(xi − xj) · ∏_{i>l}(yi − yj) · Res(U, V )

� Example 41.11.7. Let us consider

U = (X − 1)X(X − a) and V := (X + 1)(X + 2)(X − b)

and choose

x_{1} = 1, x_{2} = −1, x_{3} = −2, y_{1} = 1, y_{2} = −2, y_{3} = 2

so that

|C(x_{i}, y_{j})|
= | \begin{matrix}6(ab − a − b + 1) & −12(ab − a + 2b − 2) & 12(ab − a − 2b + 2) \\ 6(ab − a + b − 1) & 0 & 8(ab − 2a + b − 2) \\ 24(ab − a + 2b − 2) & −12(ab + 2a + 2b + 4) & 36(ab − 2a + 2b − 4) \end{matrix} |

= | \begin{matrix}6(a − 1)(b − 1) & −12(a + 2)(b − 1)) & 12(a − 2)(b − 1) \\ 6(a + 1)(b − 1) & 0 & 8(a + 1)(b − 2) \\ 24(a + 2)(b − 1) & −12(a + 2)(b + 2) & 36(a + 2)(b − 2) \end{matrix} |
= 2^{6}3^{3}(b − 1)b(a − b)(a + 1)(a + 2).

�

41.12 Toward Cardinal’s Conjecture

Cayley’s formulation of the bezoutic matrix in terms of \frac{U(X)V (Y)−V (X)U(Y)}{X−Y}

which was interpreted by Dixon in matricial terms as \frac{| \begin{matrix}U(X) & V (X) \\ U(Y) & V (Y) \end{matrix} |}{(X−Y)} has been expressed by Cardinal in different (but equivalent) ways as:

\frac{U(X)V (Y) − V (X)U(Y)}{X − Y} = | \begin{matrix}\frac{U(X)−U(Y )}{X−Y} & U(Y) \\ \frac{V (X)−V (Y )}{X−Y} & V (Y) \end{matrix} | = | \begin{matrix}\frac{U(X)−U(Y )}{X−Y} & U(X) \\ \frac{V (X)−V (Y )}{X−Y} & V (X) \end{matrix} | .

In a similar way, given a set of n polynomials

F := {f_{1}, …, f_{n}} ∈ k[X_{1}, …, X_{n}]

and denoting, for each polynomial g ∈ k[X_{1}, …, X_{n}]

• D(g, F) and D(g, F) respectively the Dixon polynomial and matrix of f_{1}, …, f_{n}, g,

• g(X_{i}) := g(Y_{1}, …, Y_{i}, X_{i+1}, …, X_{n}), for each i, 0 ≤ i ≤ n,

• δ_{i}(g) := \frac{g(X_{i})−g(X_{i−1})}{X_{i}−Y_{i}}, for each i and

• δ_{i}(g, h) := \frac{g(X_{i})h(X_{i−1})−h(X_{i})g(X_{i−1})}{X_{i}−Y_{i}}, for each i and each h ∈ k[X_{1}, …, X_{n}],

Cayley’s interpretation by Cardinal was extended by himself in order to give an alternative representation of the Dixon polynomials D(1, F) and D(X_{i}, F)

Lemma 41.12.1 (Cardinal). We have,

D(1, F) := | \begin{matrix}δ_{1}(f_{1}) & … & δ_{1}(f_{n}) \\ ⋮ & ... & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) \end{matrix} |

and, for each i, 1 ≤ i ≤ n,

D(X_{i}, F) := | \begin{matrix}δ_{1}(f_{1}) & … & δ_{1}(f_{n}) \\ ⋮ & ... & ⋮ \\ δ_{i−1}(f_{1}) & … & δ_{i−1}((f_{n}) \\ δ_{i}(X_{i}, f_{1}) & … & δ_{i}(X_{i}, f_{n}) \\ δ_{i+1}(f_{1}) & … & δ_{i+1}((f_{n}) \\ ⋮ & ... & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) \end{matrix} | .

Proof. If, in (41.6), we set f_{n+1} = 1, we subtract the (j)^{th} row to the j + 1^{th} row and divide it by X_{j} − Y_{j} for each j, 1 ≤ j ≤ n, we obtain

D(1, F) = | \begin{matrix}f_{1}(X_{1}, …, X_{n}) & … & f_{n}(X_{1}, …, X_{n}) & 1 \\ δ_{1}(f_{1}) & … & δ_{1}(f_{n},) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{i}(f_{1}) & … & δ_{i}((f_{n},) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1},) & … & δ_{n}(f_{n},) & 0 \end{matrix} |

If in (41.6), we set, instead, f_{n+1} = X_{i} and, after subtracting the (j)^{th} row to the j + 1^{th} row and dividing it by X_{j} − Y_{j} for each j, 1 ≤ j ≤ n ,we multiply the i^{th} row by Y_{i} and add to it the (i − 1)^{th} row of (41.6) since

| \begin{matrix}f_{1}(X_{0}) & … & f_{n+1}(X_{0}) \\ ⋮ & ... & ⋮ \\ f_{1}(X_{i−1}) & … & f_{n+1}(X_{i−1}) \\ f_{1}(X_{i−1}) & … & f_{n+1}(X_{i−1}) \\ f_{1}(X_{i+1}) & … & f_{n+1}(X_{i+1}) \\ ⋮ & ... & ⋮ \\ f_{1}(X_{n}) & … & f_{n+1}(X_{n}) \end{matrix} | = 0 and

Y_{i} \frac{f_{j}(X_{i}) − f_{j}(X_{i−1})}{X_{i} − Y_{i}} − f_{j}(X_{i−1}) = \frac{Y_{i}f_{j}(X_{i}) − X_{i}f_{j}(X_{i−1})}{X_{i} − Y_{i}} = δ_{i}(X_{i}, f_{j}),

we obtain

Y_{i}D(X_{i}, F) = | \begin{matrix}δ_{1}(f_{1}) & … & δ_{1}(f_{n},) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{i−1}(f_{1}) & … & δ_{i−1}((f_{n},) & 0 \\ δ_{i}(X_{i}, f_{1},) & … & δ_{i}(X_{i}, f_{n}) & 0 \\ δ_{i+1}(f_{1}) & … & δ_{i+1}((f_{n},) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1},) & … & δ_{n}(f_{n},) & 0 \\ f_{1}(Y_{1}, …, Y_{n}) & … & f_{n}(Y_{1}, …, Y_{n}) & Y_{i} \end{matrix} |

and we are through.  �

With a similar proof we also have Lemma 41.12.2 (Cardinal–Mourrain). For each polynomial g it holds

D(g, F) = \frac{| \begin{matrix}f_{1}(X_{0}) & … & f_{n}(X_{0}) & g(X_{0}) \\ f_{1}(X_{1}) & … & f_{n}(X_{1}) & g(X_{1}) \\ ⋮ & ... & ⋮ & ⋮ \\ f_{1}(X_{n}) & … & f_{n}(X_{n}) & g(X_{n}) \end{matrix} |}{∏_{j=1}^{n} X_{j} − Y_{j}}
= | \begin{matrix}f_{1}(X_{0}) & … & f_{n}(X_{0}) & g(X_{0})) \\ δ_{1}(f_{1}) & … & δ_{1}(f_{n}) & δ_{1}(g) \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) & δ_{n}(g) \end{matrix} |  (41.7)

= | \begin{matrix}δ_{1}(f_{1}) & … & δ_{1}(f_{n}) & δ_{1}(g) \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) & δ_{n}(g) \\ f_{1}(X_{n}) & … & f_{n}(X_{n}) & g(X_{n})) \end{matrix} |  (41.8)

�

Remark 41.12.3 (Becker, Cardinal et al.). Jacobi’s interpretation of the bezoutic matrix in terms of Jacobians (pg. 103) can be extended to more than two forms: with the present notation since

\frac{Y_{i}^{ν} − X_{i}^{ν}}{Y_{i} − X_{i}} = ∑_{j=0}^{ν−1} Y_{i}^{j}X_{i}^{ν−1−j}

for the polynomial

δ_{i}(h)(X_{i}, Y_{i}) = \frac{h(X_{i}) − h(X_{i−1})}{X_{i} − Y_{i}} ∈ k[Y_{1}, …, Y_{i−1}, X_{i+1}, …, X_{n}][X_{i}, Y_{i}]

we have

δ_{i}(h)(X_{i}, X_{i}) = \frac{∂h(Y_{1}, …, Y_{i−1}, X_{i}, X_{i+1}, …, X_{n}}{∂X_{i}}

so that if in D(1, F) we substitute each Y_{i} with X_{i} we obtain the Jacobian

matrix of F.  �

Given the polynomial ring P := k[X_{1}, …, X_{n}] and its monomial k-basis T we introduce n futher variables Y_{1}, …, Y_{n} and we denote P_{Y} := k[Y_{1}, …, Y_{n}], T_{Y} the monomial k-basis of P_{Y}, and P_{⊗} the ring

P_{⊗} := P ⊗ P_{Y} = k[X_{1}, …, X_{n}, Y_{1}, …, Y_{n}],

whose k basis is {τ ⊗ ω : τ ∈ T , ω ∈ T_{Y} }.

Let us consider a set of n polynomials F := {f_{1}, …, f_{n}} ∈ P generating an ideal I and denote A := P/I; with a slight abuse of notation we denote I also the ideal in P_{Y} generated by {f_{1}(Y_{1}, …, Y_{n}), …, f_{n}(Y_{1}, …, Y_{n})} and A := P_{Y} /I.

With this notation we have

A ⊗_{k} A = P_{⊗}/𝕀 (f_{i}(X_{1}, …, X_{n}), f_{i}(Y_{1}, …, Y_{n}), 1 ≤ i ≤ n) ;

in connection we also denote I_{X} := I ⊗ P_{Y} ⊂ P_{⊗} and I_{Y} := P ⊗ I ⊂ P_{⊗}.

For each g ∈ P denote D(g) := D(g, F) ∈ P_{⊗}; denote also D_{0} := D(1) and D_{i} := D(X_{i}), 1 ≤ i ≤ n.

Let 𝔞 ⊂ T and 𝔟 ⊂ T_{Y} be suitable ordered finite sets such that we can express each D_{i}, 0 ≤ i ≤ n, as

D_{i} := ∑_{τ∈𝔞} ∑_{ω∈𝔟} d_{τω}^{(i)}τ ⊗ ω = ∑_{τ∈𝔞} ∑_{ω∈𝔟} d_{τω}^{(i)}τω ∈ P_{⊗}

and denote D_{i} := (d_{τω}^{(i)}) the corresponding Dixon matrix.

Lemma 41.12.4. D(f) = 0 for each f ∈ F.  �

Lemma 41.12.5 (Cardinal). For each g ∈ P it holds

D(g) − g(X_{1}, …, X_{n})D_{0} ∈ I_{X} and D(g) − g(Y_{1}, …, Y_{n})D_{0} ∈ I_{Y} .

Proof. By expanding D(g, F) along the first (respectively: last) row of (41.7) (respectively: (41.8)) we obtain D(g) − g(X_{1}, …, X_{n})D_{0} ∈ I_{X} (respectively:

D(g) − g(Y_{1}, …, Y_{n})D_{0}).  �

Corollary 41.12.6 (Cardinal). For each g ∈ P, denoting

D(g) := ∑_{τ∈𝔞} ∑_{ω∈𝔟} d_{τω}τ ⊗ ω,

we have

∑_{τ∈𝔞} d_{τω}τ ≡ g(X_{1}, …, X_{n}) ∑_{τ∈𝔞} d_{τω}^{(0)}τ mod I_{X} for each ω ∈ 𝔟

and

∑_{ω∈𝔟} d_{τω}ω ≡ g(Y_{1}, …, Y_{n}) ∑_{ω∈𝔟} d_{τω}^{(0)}ω mod I_{Y} for each τ ∈ 𝔞.

�

Corollary 41.12.7. For each g ∈ P it holds

(g(Y_{1}, …, Y_{n}) − g(X_{1}, …, X_{n}))D_{0} ∈ 𝕀 (f_{i}(X_{0}), f_{i}(X_{n}), 1 ≤ i ≤ n) .

�

Corollary 41.12.8. For each i, 1 ≤ i ≤ n, ω ∈ 𝔟 and τ ∈ 𝔞, there are polynomials k_{ω}^{(i)}(X_{1}, …, X_{n}) ∈ P and h_{τ}^{(i)}(Y_{1}, …, Y_{n}) ∈ P_{Y} such that

(1) D_{i} − X_{i}D_{0} = | \begin{matrix}f_{1}(X_{0}) & … & f_{n}(X_{0}) & 0 \\ δ_{1}(f_{1}) & … & δ_{1}(f_{n}) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{i}(f_{1}) & … & δ_{i}(f_{n}) & 1 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) & 0 \end{matrix} | = ∑_{ω∈𝔟} k_{ω}^{(i)}ω;

(2) k_{ω}^{(i)}(X_{1}, …, X_{n}) ∈ I for each ω ∈ 𝔟;

(3) D_{i} − Y_{i}D_{0} = | \begin{matrix}δ_{1}(f_{1}) & … & δ_{1}(f_{n}) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{i}(f_{1}) & … & δ_{i}(f_{n}) & 1 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) & 0 \\ f_{1}(X_{n}) & … & f_{n}(X_{n}) & 0 \end{matrix} | = ∑_{τ∈𝔞} τh_{τ}^{(i)};

(4) h_{τ}^{(i)}(X_{1}, …, X_{n}) ∈ I for each τ ∈ 𝔞.  �

Let us now assume that that I := 𝕀(F) is an affine complete intersection (Definition 36.1.1); as a consequence we have^{71}

Fact 41.12.9. With the present notation and under the assumption that I := 𝕀(F) is an affine complete intersection, it holds:

(1) A is a finite k-dimensional vector space;

(2) the morphism D_{0}| : Hom(A, k) → A which associates to the functional Λ : A → k the element ∑_{τ∈𝔞} ∑_{ω∈𝔟} d_{τω}^{(0)}τΛ(ω) ∈ A is actually an Aisomorphism;

71 Compare B.Mourrain, Bezoutian and quotient ring structure J. Symb. Comp. 39 (2005), 397-415.

(3) Hom(A, k) is a free A-module whose basis element ℓ ∈ Hom(A, k) satisfies D_{0}|ℓ = 1;

(4) D_{0} ≡ ∑_{i=1}^{D} a_{i} ⊗ b_{i} in A ⊗ A where D = dim_{k}(A) and {a_{i}, 1 ≤ i ≤ D} and {b_{i}, 1 ≤ i ≤ D} are suitable k-bases of A;

(5) both 𝔞 and 𝔟 are k-generating sets of A.  �

41.13 Cardinal–Mourrain Algorithm

Let P, P_{Y}, P_{⊗}, T , T_{Y}, F := {f_{1}, …, f_{n}} ∈ P a set of n polynomials generating the affine complete intersection ideal I, A, I_{X}, I_{Y}, D_{i}, 0 ≤ i ≤ r, 𝔞, 𝔟, D_{i} := (d_{τω}^{(i)}) , 0 ≤ i ≤ r, be as defined in page 120.

Let us morevoer denote V := Span_{k}(𝔞), W := Span_{k}(𝔟),

K_{0} := Span_{k}{k_{ω}^{(i)} : 1 ≤ i ≤ n, ω ∈ 𝔟} ∩ V ⊂ V ∩ I ⊂ P

and

H_{0} := Span_{k}{h_{τ}^{(i)} : 1 ≤ i ≤ n, τ ∈ 𝔞} ∩ W ⊂ W ∩ I ⊂ P_{Y}

where k_{ω}^{(i)}, h_{τ}^{(i)} are the polynomials whose existence is stated in Corollary 41.12.8.

We present an algorithm, which iteratively extends the vectorspaces K_{0}, H_{0}, returning, at terminantion,

• vectorspaces K, H, K_{0} ⊆ K = I ∩ V, H_{0} ⊆ H = I ∩ W;

• the supplementary vectorspaces A, B, A ⊗ K = V, B ⊗ H = W, which satisfy the relation dim_{k}(A) = dim_{k}(B) =: δ;

• the bases a := {a_{1}, …, a_{δ}}, {b_{1}, …, b_{δ}} of A and B respectively;

• δ-square matrices M_{i}, 0 ≤ i ≤ n,

such that

(1) M_{0} is invertible,

(2) \overline{M}_{p} := M_{0}^{−1}M_{p} := (m_{ji}^{(p)}) satisfy

X_{p}a_{i} ≡ ∑_{j=1}^{δ} m_{ji}a_{j} mod I, ∀i, p, 1 ≤ i ≤ δ, 1 ≤ p ≤ n

so that, in particular

(3) A = P/I ≅ V/K ≅ Span_{k}(a) and

(4) the assignement of the k-basis a and the square matrices

\overline{M}_{p} := M([X_{p}], a), 1 ≤ p ≤ n,

is a Gröbner representation of I.

Recall that the Dixon polynomials D_{i} are decomposed with respect to the same ordered sets of polynomials 𝔞 ⊂ P and 𝔟 ⊂ P_{Y} which are indexing the rows and the columns of the Dixon matrices D_{i} so that

D_{i} = 𝔞^{T} D_{i}𝔟.

The algorithm in each step performes simultaneously the same operation on the n + 1 matrices D_{i} appying invertible transformations P, Q on their rows and columns, thus returning

P D_{0}Q, P D_{1}Q, …, P D_{n}Q

and transforming the bases of V and W so that the indexes of the common rows and columns of the matrices P D_{i}Q are respectively P^{−1}^{T} 𝔞 and Q^{−1}𝔟.

We present in Figure 41.3 the general scheme of the algorithm whose single operations we discuss here through an example.

Fig. 41.3. Cardinal–Mourrain Algorithm i := 0 Repeat

⋆ Apply the saturation step on K_{i} returning K′ : K_{i} ⊆ K′ ⊆ I ∩ V, Pertorm the quartering step on the matrices D_{i}, Pertorm the column reduction step returning K_{i+1} : K′ ⊆ K_{i+1} ⊂ I ∩ V, Pertorm the diagonalization step, Pertorm the row step returning H_{i+1}; H_{i} ⊆ H_{i+1} ⊆ I ∩ W,

until H_{i+1} = H_{i} and K_{i+1} = K_{i}.

Example 41.13.1 (Cardinal). As an example let us consider the ideal generated by F = {f_{1}, f_{2}} ∈ k[X_{1}, X_{2}] where

f_{1} = X_{1}^{2} + X_{1}X_{2}^{2} − 1, f_{2} = X_{1}^{2}X_{2} + X_{1}.

We thus have

D_{0} = −X_{1}X_{2}^{2}Y_{1} − X_{1}X_{2}Y_{1}Y_{2} − X_{2}Y_{1}^{2}Y_{2} + X_{1}Y_{1}^{2} + Y_{1}^{3} − X_{2}Y_{1} − Y_{1}Y_{2},
D_{1} = −X_{1}X_{2}^{2}Y_{1}^{2} − X_{1}X_{2}Y_{1}^{2}Y_{2} + X_{1}Y_{1}^{3} + Y_{1}^{2},
D_{2} = −X_{1}X_{2}^{2}Y_{1}Y_{2} − X_{2}^{2}Y_{1}^{2}Y_{2} + X_{1}X_{2}Y_{1}^{2} + X_{2}Y_{1}^{3} − X_{2}^{2}Y_{1}
− X_{2}Y_{1}Y_{2} − X_{1}X_{2} − X_{1}Y_{1} − X_{2}Y_{1} − 1;
D_{1} − X_{1}D_{0} = (X_{1}^{2}X_{2}^{2} + X_{1}X_{2})Y_{1} + (−X_{1}X_{2}^{2} − X_{1}^{2} + 1)Y_{1}^{2} + (X_{1}^{2}X_{2} + X_{1})Y_{1}Y_{2},
D_{2} − X_{2}D_{0} = (X_{2}^{3}X_{1} − X_{2} − X_{1})Y_{1} + (−X_{2}X_{1} − 1);
D_{1} − Y_{1}D_{0} = X_{2}(Y_{2}Y_{1}^{3} + Y_{1}^{2}) + (Y_{2}Y_{1}^{2} − Y_{1}^{4} + Y_{1}^{2}),
D_{2} − Y_{2}D_{0} = X_{2}^{2}(−Y_{2}Y_{1}^{2} − Y_{1}) + X_{2}X_{1}(Y_{2}^{2}Y_{1} + Y_{1}^{2} − 1) + X_{2}(Y_{2}^{2}Y_{1}^{2} + Y_{1}^{3} − Y_{1})
− X_{1}(Y_{2}Y_{1}^{2} + Y_{1}) + (Y_{2}^{2}Y_{1} − Y_{2}Y_{1}^{3} − 1)

whence

𝔞 = {1, X_{2}, X_{2}^{2}, X_{1}, X_{1}X_{2}, X_{1}X_{2}^{2}}, 𝔟 = {1, Y_{1}, Y_{1}Y_{2}, Y_{1}^{2}, Y_{1}^{2}Y_{2}, Y_{1}^{3}},
K_{0} = {X_{2}X_{1} + 1} and H_{0} = {Y_{2}Y_{1}^{2} + Y_{1}}.  �

Saturation step It consists in replacing K_{i} with K′ := K_{i}^{+} ∩ V where, for a vectorspace K, K^{+} indicates the vectorspace

K^{+} := {v_{0} + ∑_{i=1}^{n} X_{i}v_{i}, v_{i} ∈ K, 0 ≤ i ≤ n}

and the notation K^{[p]} means p iterations of the operator ·^{+}, staring from K so that K^{[p]} = (K^{[p−1]})^{+}.

Definition 41.13.2 (Mourrain). A vectorspace V ⊂ P is said to be connected to e ∈ V if, denoting E := Span_{k}{e}, for each v ∈ V \ E, there exists l > 0 such that v ∈ E^{[l]} and v = v_{0} + ∑_{i=1}^{n} X_{i}v_{i}, v_{i} with

v_{i} ∈ E^{[l−1]} ∩ V, 0 ≤ i ≤ n.  �

Example 41.13.3 (cont.). We have

K′ := {X_{2}X_{1} + 1, X_{2}(X_{2}X_{1} + 1)}.

�

Quartering step Set

d := dim_{k}(V) = dim_{k}(W), D := d − dim_{k}(K′),

choose a basis {a_{1}, …, a_{D}, a_{D+1}, …, a_{d}} of V such that {a_{D+1}, …, a_{d}} is a basis of K′ and set A := Span_{k}{a_{1}, …, a_{D}}.

Let L(K′) = {Λ ∈ Hom(P, k) : Λ(k) = 0 for each k ∈ K′} and

B := {Λ|D_{0} : Λ ∈ L(K′)} ⊃ {Λ|D_{0} : Λ ∈ L(I)}

where |D_{0} : Hom(P, k) → P_{Y} is the morphism which associates to the functional Λ : P → k the element ∑_{τ∈𝔞} ∑_{ω∈𝔟} d_{τω}^{(0)}Λ(τ)ω ∈ P_{Y}.

Choose a basis {b_{1}, …, b_{r}, b_{r+1}, …, b_{d}} of W such that {b_{1}, …, b_{r}} is a basis of B; denote H′ := Span_{k}{b_{r+1}, …, b_{d}}.

Remark that since K′ ⊂ I, B ⊃ L(I) is a generating set of A so that H′ could be chosen as a subset of I.

Based on the decompositions V = A ⊗ K′ and W = B′ ⊗ H′ the matrices D_{i} can be decomposed quarterly as

D_{0} = ( \begin{matrix}M_{0} & 0 \\ H_{0} & L_{0} \end{matrix} ) , D_{i} = ( \begin{matrix}M_{i} & K_{i} \\ H_{i} & L_{i} \end{matrix} ) , 1 ≤ i ≤ n.

Example 41.13.3 (cont.). We set D = 4 and

a_{1} = 1, a_{2} = X_{2}, a_{3} = X_{2}^{2}, a_{4} = X_{1}, a_{5} = X_{1}X_{2} + 1, a_{6} = X_{1}X_{2}^{2} + X_{2},

so that D_{0} = a_{1}Y_{1}^{3} − a_{2}Y_{1}^{2}Y_{2} + a_{4}Y_{1}^{2} − a_{5}Y_{1}Y_{2} − a_{6}Y_{1}.

We thus get

b_{1} = Y_{1}^{3}, b_{2} = Y_{1}^{2}Y_{2}, b_{3} = Y_{1}^{2}, b_{4} = 1, b_{5} = Y_{1}Y_{2}, b_{6} = Y_{1} + Y_{1}^{2}Y_{2}, and

\begin{matrix}D_{0} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 1 & 0 & 0 & 0 & 0 & 0 \\ a_{2} & 0 & −1 & 0 & 0 & 0 & 0 \\ a_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{4} & 0 & 0 & 1 & 0 & 0 & 0 \\ a_{5} & 0 & 0 & 0 & 0 & −1 & 0 \\ a_{6} & 0 & 1 & 0 & 0 & 0 & −1 \end{matrix}

\begin{matrix}D_{1} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 0 & 1 & 1 & 0 & 0 & 0 \\ a_{2} & 0 & 0 & 1 & 0 & 0 & 0 \\ a_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{4} & 1 & 0 & 0 & 0 & 0 & 0 \\ a_{5} & 0 & −1 & 0 & 0 & 0 & 0 \\ a_{6} & 0 & 0 & −1 & 0 & 0 & 0 \end{matrix}

\begin{matrix}D_{2} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 0 & 0 & −1 & 0 & 0 & 0 \\ a_{2} & 1 & 1 & 0 & 0 & 0 & −1 \\ a_{3} & 0 & 0 & 0 & 0 & 0 & −1 \\ a_{4} & 0 & 1 & 0 & 0 & 0 & −1 \\ a_{5} & 0 & 0 & 1 & −1 & 0 & 0 \\ a_{6} & 0 & 0 & 0 & 0 & −1 & 0 \end{matrix}

�

Column reduction step By Lemma 41.12.5 if a column of D_{0} represents a polynomial f ∈ P, the corresponding column of D_{i} represents X_{i}f mod I so that we can deduce that the columns of ( \begin{matrix}K_{i} \\ L_{i} \end{matrix} ) and actually of K_{i} give elements in I allowing to extend K′ returning a vectorspace K_{i+1} : K′ ⊆ K_{i+1} ⊂ I ∩ V. Example 41.13.3 (cont.). The sixth column of D_{2} returns X_{2}^{2}+X_{2}+X_{1}. � Diagonalization step By construction the number r = dim(B) of columns of M_{0} is equal to its rank; so there is an r × D matrices M_{0}^{⋆} such that M_{0}^{⋆}M_{0} = Id_{r}.

We thus multiply each D_{i} by the matrix P := ( \begin{matrix}Id_{D} & 0 \\ −H_{0}M_{0}^{⋆} & Id_{d−D} \end{matrix} ) obtaining the following decompositions

D_{0} = ( \begin{matrix}M_{0} & 0 \\ 0 & L_{0} \end{matrix} ) , D_{i} = ( \begin{matrix}M_{i} & K_{i} \\ H′_{i} & L′_{i} \end{matrix} ) , 1 ≤ i ≤ n.

This corresponds to a change of the basis a := {a_{1}, …, a_{d}} of V which becomes P^{−1}^{T} a.

Example 41.13.3 (cont.). We have M_{0}^{⋆} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) and we have

to multiply by the matrix ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 \end{matrix} ) returning with

a_{1} = 1, a_{2} = −X_{1}X_{2}^{2}, a_{3} = X_{2}^{2}, a_{4} = X_{1}, a_{5} = X_{1}X_{2} + 1, a_{6} = X_{1}X_{2}^{2} + X_{2},

\begin{matrix}D_{0} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 1 & 0 & 0 & 0 & 0 & 0 \\ a_{2} & 0 & −1 & 0 & 0 & 0 & 0 \\ a_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{4} & 0 & 0 & 1 & 0 & 0 & 0 \\ a_{5} & 0 & 0 & 0 & 0 & −1 & 0 \\ a_{6} & 0 & 0 & 0 & 0 & 0 & −1 \end{matrix}

\begin{matrix}D_{1} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 0 & 1 & 1 & 0 & 0 & 0 \\ a_{2} & 0 & 0 & 1 & 0 & 0 & 0 \\ a_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{4} & 1 & 0 & 0 & 0 & 0 & 0 \\ a_{5} & 0 & −1 & 0 & 0 & 0 & 0 \\ a_{6} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

\begin{matrix}D_{2} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 0 & 0 & −1 & 0 & 0 & 0 \\ a_{2} & 1 & 1 & 0 & 0 & 0 & −1 \\ a_{3} & 0 & 0 & 0 & 0 & 0 & −1 \\ a_{4} & 0 & 1 & 0 & 0 & 0 & −1 \\ a_{5} & 0 & 0 & 1 & −1 & 0 & 0 \\ a_{6} & 1 & 1 & 0 & 0 & −1 & −1 \end{matrix}

�

Row step We perform on the rows the same construction we have performed on the columns, thus enlarging H_{i} to H_{i+1} : H_{i} ⊆ H_{i+1} ⊆ I ∩ W. Example 41.13.3 (cont.). The sixth row of D_{2} returns −Y_{2}Y_{1} +Y_{1}^{3} −Y_{1}. � Lemma 41.13.4 (Mourrain). At the end of the algorithm denoting

\overline{K} := K_{i+1} ⊂ I ∩ V, \overline{H} := H_{i+1} ⊂ I ∩ W, A ⊂ V and B ⊂ W, the vectorspaces such that \overline{K} ⊗ A = V, \overline{H} ⊗ B = W, d := dim_{k}(V) = dim_{k}(W), δ := d − dim_{k}(\overline{K}), then it holds \overline{K}^{+} ∩ V = \overline{K} and there exist linearly independent polynomials

\{a_{1}, …, a_{δ}, a_{δ+1}, …, a_{d}\} ⊂ V, a_{i} ∈ \overline{K}, δ < i ≤ d

and a basis \{b_{1}, …, b_{δ}, b_{δ+1}, …, b_{d}\} of W with b_{i} ∈ \overline{H}, δ < i ≤ d, with respect to which the Dixon matrices D_{i} have the decompositions

D_{0} = ( \begin{matrix}Id_{δ} & 0 \\ 0 & L_{0} \end{matrix} ) , D_{i} = ( \begin{matrix}\overline{M}_{i} & 0 \\ 0 & L_{i} \end{matrix} ) , 1 ≤ i ≤ n.

Proof. At the end of the algorithm^{72},

• the saturation step does not increase \overline{K} so that \overline{K}^{+} ∩ V = \overline{K};

• the quartering step returns the decompositions

D_{0} = ( \begin{matrix}M_{0} & 0 \\ H_{0} & L_{0} \end{matrix} ) , D_{i} = ( \begin{matrix}M_{i} & K_{i} \\ H_{i} & L_{i} \end{matrix} ) , 1 ≤ i ≤ n;

• since the column reduction step does not increase \overline{K}, this means that K_{i} = 0 for each i;

• a variation of the diagonalization step in which we left-multiply by

( \begin{matrix}M_{0}^{−1} & 0 \\ −H_{0}M_{0}^{−1} & Id_{δ} \end{matrix} )

returns the decompositions

D_{0} = ( \begin{matrix}Id_{δ} & 0 \\ 0 & L_{0} \end{matrix} ) , D_{i} = ( \begin{matrix}\overline{M}_{i} & 0 \\ H′_{i} & L_{i} \end{matrix} ) , 1 ≤ i ≤ n,

with respect to a suitable basis;

• since the row step does not increase \overline{H}, this means that H′_{i} = 0 for each i

thus completing the argument.  �

Example 41.13.1 (cont.). We now have

K = {X_{1}X_{2} + 1, X_{1}X_{2}^{2} + X_{2}, X_{2}^{2} + X_{2} + X_{1}}

and H = {Y_{1} + Y_{1}^{2}Y_{2}, −Y_{2}Y_{1} + Y_{1}^{3} − Y_{1}} and we thus choose as basis for V and W respectively

a_{1} = 1, a_{2} = −X_{1}X_{2}^{2}, a_{3} = X_{1}, a_{4} = X_{2}^{2}+X_{2}+X_{1}, a_{5} = X_{1}X_{2}+1, a_{6} = X_{1}X_{2}^{2}+X_{2},

and

b_{1} = Y_{1}^{3}, b_{2} = Y_{1}^{2}Y_{2}, b_{3} = Y_{1}^{2}, b_{4} = 1, b_{5} = −Y_{2}Y_{1} + Y_{1}^{3} − Y_{1}b_{6} = Y_{1} + Y_{1}^{2}Y_{2};

72 Meening: in the last Repeet-loop at whose end K_{i+1} and H_{i+1} are not increased.

with respect to these bases we have

\begin{matrix}D_{0} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 1 & 0 & 0 & 0 & 0 & 0 \\ a_{2} & 0 & −1 & 0 & 0 & 0 & 0 \\ a_{3} & 0 & 0 & 1 & 0 & 0 & 0 \\ a_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{5} & −1 & −1 & 0 & 0 & 1 & 1 \\ a_{6} & 0 & 0 & 0 & 0 & 0 & −1 \end{matrix}

\begin{matrix}D_{1} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 0 & 1 & 1 & 0 & 0 & 0 \\ a_{2} & 0 & 0 & 1 & 0 & 0 & 0 \\ a_{3} & 1 & 0 & 0 & 0 & 0 & 0 \\ a_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{5} & 0 & −1 & 0 & 0 & 0 & 0 \\ a_{6} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

\begin{matrix}D_{2} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 0 & 0 & −1 & 0 & 0 & 0 \\ a_{2} & 1 & 1 & 0 & 0 & 0 & 0 \\ a_{3} & 0 & 1 & 0 & 0 & 0 & 0 \\ a_{4} & 0 & 0 & 0 & 0 & 0 & −1 \\ a_{5} & 0 & 0 & 1 & −1 & 0 & 0 \\ a_{6} & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix}

The next loop will prove that we have already reached the required solution since neither K nor H are enalarged and can be also used to illustrate the claim of the Lemma.

K is not enlarged neither by the saturation step nor by the column reduction step and we now left-multiply the D_{i} by

( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & −1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

obtaining the basis

\begin{aligned}a_{1} & = & −X_{1}X_{2}, & a_{2} & = & X_{1}X_{2}^{2} − X_{1}X_{2} − 1, \\ a_{3} & = & X_{1}, & a_{4} & = & X_{2}^{2} + X_{2} + X_{1}, \\ a_{5} & = & X_{1}X_{2} + 1, & a_{6} & = & X_{1}X_{2}^{2} + X_{2} \end{aligned}

and the matrices

\begin{matrix}D_{0} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 1 & 0 & 0 & 0 & 0 & 0 \\ a_{2} & 0 & 1 & 0 & 0 & 0 & 0 \\ a_{3} & 0 & 0 & 1 & 0 & 0 & 0 \\ a_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{5} & 0 & 0 & 0 & 0 & 1 & 1 \\ a_{6} & 0 & 0 & 0 & 0 & 0 & −1 \end{matrix}

\begin{matrix}D_{1} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 0 & 1 & 1 & 0 & 0 & 0 \\ a_{2} & 0 & 0 & −1 & 0 & 0 & 0 \\ a_{3} & 1 & 0 & 0 & 0 & 0 & 0 \\ a_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{5} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{6} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

\begin{matrix}D_{2} & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} \\ a_{1} & 0 & 0 & −1 & 0 & 0 & 0 \\ a_{2} & −1 & −1 & 0 & 0 & 0 & 0 \\ a_{3} & 0 & 1 & 0 & 0 & 0 & 0 \\ a_{4} & 0 & 0 & 0 & 0 & 0 & −1 \\ a_{5} & −1 & −1 & 0 & −1 & 0 & 0 \\ a_{6} & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} .

�

Claim 41.13.4 (Cardinal–Mourrain). The δ-square matrices \overline{M}_{i}^{T} (respectively \overline{M}_{i}) are the matrices M([X_{i}], a) (respectively M([Y_{i}], b)) of multiplication by the variables X_{i} (resp. Y_{i}) in the basis a := {a_{1}, …, a_{δ}} (resp.

b := {b_{1}, …, b_{δ}}) of A.  �

Historical Remark 41.13.5. A preliminary version of the algorithm, without the saturation step, was proposed in his these by Cardinal in 1993 which conjectured the claim above.

The insertion of the saturation step and the proof of the claim, under the further assumption that V is connected to 1, is due to Mourrain in 2003.

As regard the saturation step Mourrain^{73} comments

The reason why we need to introduce this saturation step is that if we multiply all the [Dixon polynomials] by an element of the form 1 + fg, f ∈ P, g ∈ P_{Y} with f, g conveniently chosen, we could obtain

matrices of the form ( \begin{matrix}D_{i} & 0 \\ 0 & D_{i} \end{matrix} ) . Applying only the […] steps as

described by Cardinal, would not allow us to avoid the duplication

73 B.Mourrain, Bezoutian and quotient ring structure J. Symb. Comp. 39 (2005), 397-415

of the structure of A. Moreover, if f and g are in I, the polynomials (1 + fg)D_{i} share the same properties, modulo I, as the [Dixon polynomials] D_{i}. To handle this problem, we add the saturation step, which will “connect” the two blocks, provided that the vector space V is connected to an element e. This is the hypothesis that will be made hereafter to prove the main theorem. This hypothesis is easy to check in practice, and usually we have e = 1. Moreover, it is satisfied when the polynomials f_{i} are monomials. We do not have a proof that this extends by linearity to any polynomial f_{i}. […] To simplify the proof, we will assume hereafter that e = 1. The proof can be extended to any e, by showing that, in this case, e is invertible

in A^{74} and by dividing by e.  �

Example 41.13.1 (cont.). Gauss-reducing {a_{1}, a_{2}, a_{3}} and {b_{1}, b_{2}, b_{3}} with respect the basis elements of, respectively, K and H we can set

a′_{1} := a_{5} + a_{1} = 1, a′_{2} := a_{2} − a_{6} + a_{5} = −X_{2}, a′_{3} := X_{1}

and

b′_{1} := Y_{1}^{3}, b′_{2} := b_{2} − b_{6} = −Y_{1}, b′_{3} = Y_{1}^{2}

and we have^{75}

\overline{M}_{1} = ( \begin{matrix}0 & 1 & 1 \\ 0 & 0 & −1 \\ 1 & 0 & 0 \end{matrix} ) and \overline{M}_{2} = ( \begin{matrix}0 & 0 & −1 \\ −1 & −1 & 0 \\ 0 & 1 & 0 \end{matrix} ) .

�

74 By assumption we have A = Span_{k}(a) and a_{i} = ea′_{i} for some a′_{i} so that

A ∋ 1 = ∑_{i} c_{i}a_{i} = (∑_{i} c_{i}a′_{i}) e.

75 The reader can check the result using the deglex Gröbner basis of I induced by X_{1} < X_{2} which is

{X_{1}X_{2} + 1, X_{1}^{2} − X_{2} − 1, X_{2}^{2} + X_{2} + X_{1}}

so that, in particular, Y_{1}^{3} ≡ Y_{1} − 1 and Y_{1}^{2} ≡ Y_{2} + 1.

41.14 Mourrain: Proving Cardinal’s Conjecture

Let us use the same notation as in the last sections; in particular we have a := {a_{1}, …, a_{δ}}, b := {b_{1}, …, b_{δ}} A = Span_{k}(a), B = Span_{k}(b) and we set \overline{M}_{p} := M_{0}^{−1}M_{p} := (m_{ji}^{(p)}).

Proposition 41.14.1 (Mourrain). It holds

(1) X_{p}a_{i} = ∑_{l=1}^{δ} m_{li}^{(p)}a_{l} − κ_{i}^{(p)}, κ_{i}^{(p)} ∈ \overline{K} for each p, i, 1 ≤ p ≤ n, 1 ≤ i ≤ δ;

(2) Y_{p}b_{i} = ∑_{l=1}^{δ} m_{il}^{(p)}b_{l} − σ_{i}^{(p)}, σ_{i}^{(p)} ∈ \overline{H} for each p, i, 1 ≤ p ≤ n, 1 ≤ i ≤ δ;

(3) D(X_{p}X_{q}) = X_{q}D(X_{p}) + Y_{p} (D(X_{q}) − X_{q}D(1)) for p < q;

(4) D(X_{p}X_{q}) = Y_{p}D(X_{q}) + X_{q} (D(X_{p}) − Y_{p}D(1)) for p < q;

(5) D(X_{p}X_{q}) = ∑_{1≤i,j,l≤δ} m_{li}^{(q)}m_{ij}^{(p)}a_{l} ⊗ b_{j} + X_{q}χ_{1}^{(p,q)} + Y_{p}χ_{2}^{(p,q)} + χ_{3}^{(p,q)} for

p < q and suitable elements, χ_{1}^{(p,q)}, χ_{2}^{(p,q)}, χ_{3}^{(p,q)} ∈ \overline{K} ⊗ \overline{K};

(6) D(X_{p}X_{q}) = ∑_{1≤i,j,l≤δ} m_{li}^{(p)}m_{ij}^{(q)}a_{l} ⊗ b_{j} + X_{p}χ_{4}^{(p,q)} + Y_{q}χ_{5}^{(p,q)} + χ_{6}^{(p,q)} for

p < q and suitable elements χ_{4}^{(p,q)}, χ_{5}^{(p,q)}, χ_{6}^{(p,q)} ∈ \overline{K} ⊗ \overline{K}.

Proof.

(1) We have

D_{p} = X_{p}D_{0} + ∑_{i=1}^{δ} κ_{i}^{(p)}b_{i} + ∑_{l=δ+1}^{d} κ_{l}^{(p)}b_{l}, κ_{i}^{(p)}, κ_{l}^{(p)} ∈ K_{0}.

By identifying the coefficients of each b_{i}, 1 ≤ i ≤ δ we have

∑_{l=1}^{δ} m_{li}^{(p)}a_{l} = X_{p}a_{i} + κ_{i}^{(p)}, κ_{i}^{(p)} ∈ K_{0} ⊂ \overline{K}.

(2) Similar proof as (1).

(3) We have

D(X_{p}X_{q}) = | \begin{matrix}f_{1}(X_{1}, …, X_{n}) & … & f_{n}(X_{1}, …, X_{n}) & X_{p}X_{q} \\ δ_{1}(f_{1}) & … & δ_{1}(f_{n}) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{p}(f_{1}) & … & δ_{p}(f_{n}) & X_{q} \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{q}(f_{1}) & … & δ_{q}(f_{n}) & Y_{p} \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) & 0 \end{matrix} |

= X_{q} | \begin{matrix}f_{1}(X_{1}, …, X_{n}) & … & f_{n}(X_{1}, …, X_{n}) & X_{p} \\ δ_{1}(f_{1}) & … & δ_{1}(f_{n}) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{p}(f_{1}) & … & δ_{p}(f_{n}) & 1 \\ ⋮ & ... & ⋮ & ⋮ \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) & 0 \end{matrix} |

+ Y_{p} | \begin{matrix}f_{1}(X_{1}, …, X_{n}) & … & f_{n}(X_{1}, …, X_{n}) & 0 \\ δ_{1}(f_{1}) & … & δ_{1}(f_{n}) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{q}(f_{1}) & … & δ_{q}(f_{n}) & 1 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) & 0 \end{matrix} | .

(4) Similarly as (3), we have to develop

D(X_{p}X_{q}) = | \begin{matrix}δ_{1}(f_{1}) & … & δ_{1}(f_{n}) & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{p}(f_{1}) & … & δ_{p}(f_{n}) & X_{q} \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{q}(f_{1}) & … & δ_{q}(f_{n}) & Y_{p} \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n}(f_{1}) & … & δ_{n}(f_{n}) & 0 \\ f_{1}(Y_{1}, …, y_{n}) & … & f_{n}(Y_{1}, …, Y_{n}) & Y_{p}Y_{q} \end{matrix} | .

(5) By (2) we have

D(X_{p}X_{q}) = X_{q}D(X_{p}) + Y_{p} (D(X_{q}) − X_{q}D(1))
= ∑_{1≤i,j≤δ} m_{ij}^{(p)}X_{q}a_{i} ⊗ b_{j} + X_{q}χ_{1} + Y_{p} ∑_{1≤i≤δ} κ_{i}^{(q)} ⊗ b_{i} + Y_{p}χ_{2}

= ∑_{1≤i,j≤δ} m_{ij}^{(p)} ( ∑_{1≤l≤δ} m_{li}^{(q)}a_{l} − κ_{i}^{(q)}) ⊗ b_{j} + X_{q}χ_{1}

+ ∑_{1≤i≤δ} κ_{i}^{(q)} ⊗ ( ∑_{1≤l≤δ} m_{il}^{(p)}b_{l} − σ_{i}^{(p)}) + Y_{p}χ_{2}

= ∑_{1≤i,j,l≤δ} m_{li}^{(q)}m_{ij}^{(p)}a_{l} ⊗ b_{j} − ∑_{1≤i,j≤δ} m_{ij}^{(p)}κ_{i}^{(q)} ⊗ b_{j}

+ ∑_{1≤i,l≤δ} m_{il}^{(p)}κ_{i}^{(q)} ⊗ b_{l} + X_{q}χ_{1} + Y_{p}χ_{2} + χ_{3}
= ∑_{1≤i,j,l≤δ} m_{li}^{(q)}m_{ij}^{(p)}a_{l} ⊗ b_{j} + X_{q}χ_{1} + Y_{p}χ_{2} + χ_{3}.

(6) Similarly as (5).  �

Corollary 41.14.2 (Mourrain). The matrices \overline{M}_{p} = (m_{ji}^{(p)}) commute.

�

With a slight abuse of notation we will also denote \overline{M}_{p} the map

\overline{M}_{p} : A → A, a_{i} ↦ ∑_{l=1}^{δ} m_{ml}^{(p)}a_{l}

which corresponds to the multiplication by X_{p} modulo \overline{K}.

Since these operations commute, for each f(X_{1}, …, X_{n}) ∈ P we define

f(\overline{M}) := f(\overline{M}_{1}, …, \overline{M}_{n}) : A → A

and N(f) = f(\overline{M})(1) so that N is a map N : P → A.

Proposition 41.14.3 (Mourrain). If V is connected to 1, the ideal H := 𝕀(\overline{K}) ⊂ P generated by \overline{K} satisfies H = I.

Proof. By construction we have H ⊂ I.

The assumption that V = A⊗\overline{K} is connected to 1 implies the existence of c_{1}, …, c_{δ} ∈ k, κ ∈ \overline{K} such that u = ∑_{i=1}^{δ} c_{i}a_{i} = 1 − κ, so that, in particular, u is invertible in A and there exists Λ ∈ Hom(P, k) such that D_{0}|Λ = u.

By expanding (41.7) along the last column, we have 0 = D(f_{i}) = f_{i}(X_{1}, …, X_{n})D_{0} + ∑_{j=1}^{n}(−1)^{j}δ_{j}(f_{i})E_{j} for suitable E_{j} ∈ \overline{K} ⊗ P_{Y}.

Thus, for each i,

f_{i} = f_{i}κ + f_{i}u = f_{i}κ + f_{i}D_{0}|Λ = f_{i}κ − ∑_{j=1}^{n}(−1)^{j}δ_{j}(f_{i})E_{j}|Λ ∈ H.

�

Proposition 41.14.4 (Mourrain). Assume V is connected to 1 and 1 ∉ \overline{K}. Then

(1) for each f ∈ V, f − f(\overline{M})(1) ∈ \overline{K};

(2) for each a ∈ A, N(a) = a and N(\overline{K}) = \{0\};

(3) f − f(\overline{M})(1) ∈ H for each f ∈ P;

(4) ker(N) = H.

Proof.

(1) Assume that for each g ∈ Span_{k}\{1\}^{[l−1]}∩V we have g−g(\overline{M})(1) ∈ \overline{K} and let f ∈ Span_{k}{1}^{[l]}∩V. Since V is connected to 1, we have f = ∑_{i=1}^{s} X_{l_{i}}g_{i} with 1 ≤ l_{i} ≤ n and g_{i} ∈ Span_{k}{1}^{[l−1]} ∩ V. Thus

f − f(\overline{M})(1) = ∑_{i=1}^{s} X_{l_{i}} (g_{i} − g_{i}(\overline{M})(1)) + (X_{l_{i}}g_{i}(\overline{M})(1) − \overline{M}_{l_{i}}g_{i}(\overline{M})(1) .

We have g_{i} − g_{i}(\overline{M})(1) ∈ \overline{K} by induction assumption and X_{l_{i}}g_{i}(\overline{M})(1) − \overline{M}_{l_{i}}g_{i}(\overline{M})(1) ∈ \overline{K} by Lemma 41.12.5. Therefore f −f(\overline{M})(1) ∈ \overline{K}^{+}∩V = \overline{K}, the last equality being due to the saturation step. Since the induction hypothesis is true for f = 1, then the claim follows.

(2) For any polynomial a ∈ A and any polynomial κ ∈ \overline{K} we have • a − a(\overline{M})(1) ∈ \overline{K} ∩ A = \{0\} and

• κ(\overline{M})(1) = κ − (κ − κ(\overline{M})(1)) ∈ \overline{K} ∩ A = \{0\}

which implies that, for each a ∈ A, N(a) = a and N(\overline{K}) = \{0\}.

(3) Just a few slight adapations allow to use the same argument^{76} used for (1) to prove, again by induction, that f − f(\overline{M})(1) ∈ H for each f ∈ P.

(4) Since N(\overline{K}) = \{0\}, H ⊂ ker(N). Conversely, for each f ∈ ker(N),

f = f − N(f) = f − f(\overline{M})(1) ∈ H;
thus ker(N) ⊂ H.  �

Theorem 41.14.5 (Mourrain). Assume V is connected to 1. The δ-square matrices \overline{M}_{i}^{T} (respectively \overline{M}_{i}) are the matrices M([X_{i}], a) (respectively M([Y_{i}], b)) of multiplication by the variables X_{i} (resp. Y_{i}) in the basis a := {a_{1}, …, a_{δ}} (resp. b := {b_{1}, …, b_{δ}}) of A.

Proof. If 1 ∈ \overline{K} ⊂ I, then A = \{0\} and, by the saturation step, \overline{K} = V and the claim is trivial.

Since V is connected to 1, if 1 ∉ \overline{K} we may assume 1 to be an element of the basis A. Then by the proposition above ker(N) = H = I and Im(N) = A

so that A ≅ P/I = A.  �

41.15 Mourrain: A Gröbner-free Solver

Remark 41.15.1. If V is connected to 1, Cardinal–Moirrain Algorithm thus returns a Gröbner representation a, \overline{M}_{p}, 1 ≤ p ≤ n, of I.

These data are the ones required by Auzinger–Stetter Algorithm (Com-

pare Section 40.8).  �

76 P is connected to 1; the claim holds for f = 1; g_{i} − g_{i}(\overline{M})(1) ∈ H by induction; X_{l_{i}}g_{i}(\overline{M})(1) − \overline{M}_{l_{i}}g_{i}(\overline{M})(1) ∈ I by Lemma 41.12.5.

Remark 41.15.2 (Mourrain). Let us now set d_{i} := deg(f_{i}), D := max_{i}{d_{i}} and d := 1 + ∑_{i=1}^{n}(d_{i} − 1).

Denote ν a bound of the size of the matrices D_{i}, which is at most the number of terms of degree bounded by d, that is, by Stirling’s formula, O(e^{n}D^{n}).

If V is connected to 1, a Gröbner representation a, \overline{M}_{p}, 1 ≤ p ≤ n, of I where the basis elements a_{i} ∈ a satisfy deg(a_{i}) ≤ d, can be computed in O(nν^{4}) arithmetical operations.

In fact Cardinal–Moirrain Algorithm requires to perform at most ν loops

each performing linear transformations over n matrices of size ν.  �

Cardinal–Moirrain Algorithm can also efficiently substitute Buchberger’s Algorithm to provide a good complexity procedure to solve the membership test.

Proposition 41.15.3 (Mourrain). For each f ∈ P it is possible to test whether f ∈ I in O(nν^{4}L) where L denotes the cost of evaluating g(\overline{M}) for g ∈ F ∪ {f}.

Moreover, denoting

d̄:= deg(f) + ∑_{i=1}^{n}(d_{i} − 1), D̄ := max{deg(f_{i}), deg(f)} and ν̄ = O(e^{n}D̄^{n})

with complexity O(nν̄^{4}) it is possible to decide whether f ∈ I and, if this is the case, to produce a representation

f · u = ∑_{i=1}^{n} f_{i}g_{i} : u, g_{i} ∈ P, u ≡ 1 mod I, deg(u) ≤ d, deg(g_{i}) ≤ d̄.

Proof. Adapt Cardinal–Mourrain’s Algorithm (Figure 41.3) by substituting the saturation step with the inclusion in K_{i} and H_{i} of the n polynomials corresponding to the non-zero columns (respectively: rows) of the matrices f_{i}(\overline{M}), 1 ≤ i ≤ n; the effect, even if V is not connected to 1, is that f_{i}(\overline{M}) = 0, 1 ≤ i ≤ n.

Thus if we define σ the map

σ : P → k^{δ×δ}, f ↦ σ(f) = f(\overline{M}),

we have I ⊂ ker(σ) since, by construction f_{i}(\overline{M}) = 0, 1 ≤ i ≤ n.

On the other hand, denoting u ∈ V ⊂ P the element such that^{77} u = D_{0}|ℓ ≡ 1 mod I we have, for f ∈ ker(σ) ⊂ P

f(X_{1}, …, X_{n}) = f(X_{1}, …, X_{n}) − f(\overline{M})(u) ∈ I.

Thus we have I = ker(σ) and f ∈ I ⟺ f(\overline{M}) = 0.

77 The existence of such u, if V is connected to 1 is granted by Proposition 41.14.3 but even without this assumption is a consequence of the fact that D_{0}| is an isomorphism being F a complete interesction.

The modified algorithm requires to perform at most ν loops each performing linear transformations over n matrices of size ν and evaluations of matrices f_{i}(\overline{M}) thus its complexity is O(nν^{4}L).

Let us now modify Cardinal–Mourrain’s Algorithm (Figure 41.3) in a different way: namely we consider also the matrix D(f) and apply the modifications performed by the algorithm not only on the D_{i} but also on D(f). The effect is that we obtain, not only a basis a of A and with respect to it the matrices M([X_{i}], a) representing the multiplication by the variables, but also the matrix M_{f} := M([f], a) representing the multiplication by f. Such algorithm has complexity O(nν̄^{4}) where ν̄ = O(e^{n}D̄^{n}) and D̄ := max{deg(f_{i}), deg(f)}.

We thus have f ∈ I ⟺ M_{f} = 0 and if we expand D(f) along the first row we have

D(f) = f(X_{1}, …, X_{n})D_{0} − ∑_{i=1}^{n} f_{i}E_{i}

for suitable E_{i} ∈ P_{⊗}

Lemma 41.12.4 implies D(f)|Λ = 0 for each f ∈ I and Λ ∈ L(I) so that we have

f · u = f(X_{1}, …, X_{n})D_{0}|ℓ = ∑_{i=1}^{n} f_{i}E_{i}|ℓ

where g_{i} := E_{i}|ℓ ∈ P and u ∈ P satisfies u = D_{0}|ℓ ≡ 1 mod I and is thus invertible in I.

The degree bound is obvious by construction.  �

42. Lazard II

The introduction of Gröbner basis in the computer algebra community activated a new interest toward some older bases like Macaulay’s (Sections 23.5 and 23.6) and his algorithms (see Chapter 30), Hironaka’s standard bases (Sections 24.5-8) and Ritt’s characteristic sets.

Ritt’s results (dated 1932), strongly influenced by Noether’s results on the Decomposition Theorem, were aimed to give an algebraic standpoint to differential equations, but, as it was already usual for the Riquier’s followers (Delassus, Janet, Gunther) he translated his results also in the algebraic varaiety setting where he gave an effective decomposition algorithm which, through the further application of univiariate factorization, returned an irredundant prime decomposition of a radical ideal.

While the computer algebra community became aware of Buchberger’s result, in China Wu Wen-tsün was applying a weaker (but sufficient for his aims) version of Ritt’s algorithm as a tool toward a “mechanization” of theoremproving in elementary geometry; Wu’s version of Ritt’s result omit the hard and useless (for his aims) factorization step, thus returning a decomposition of a radical ideal into unmixed ideals.

In the Early Nineties, within the PoSSo frame, Lazard, which is the stronger expaunder and developer of the Kronecker–Duval Philosophy, reformulated Ritt’s solver avoiding the required factorization by means of Duval’s splitting via his Theorem 11.3.2 thus producing a decomposition into radical unmixed ideals, each defined via a triangular set, id est what we called (in Definition 11.4.1) a Duval admissible sequence.

Later, Möller proposed an algorithm which applies only to zero-dimensio-nal ideals, decomposing them into ideals presented through a triangular set; the theory is based on ideas related to Gianni-Kalkbrener’s Theorem and the algorithm is an adaptation of Traverso’s Algorithm 29.3.8.

Once a zero-dimensional ideal is represented through a triangular set, Kronecker–Duval Philosophy requires to transform this data into a form suitable for the computation with arihmetical expressions of its roots; suitable representations of such roots are available in older literature, for instance Gröbner’s algemaine representation and Kronecker’s parametrization; as it was proved by Alonso et al. Kronecker’s idea is more suitable than Gröbner’s since it gives a representation with lesser bit-size complexity.

An efficient algorithm to deduce, from a triangular set, a Kronecker’s parametrization or Rational Universal Representation (RUR) via suitable computation of matrix traces due to Rouillier crowns Kronecker–Duval Philosophy.

After having presented Ritt’s decomposition theory (Section 42.1 and 42.2) and the corralted solvers prosed by Ritt and Wu (Section 42.3), I discuss Lazard’s reformulation and expension of triangular sets (Section 42.4 and 42.5), the related solver (Section 42.6) and the relation between Lazard’s triangular sets and Gröbner bases (Section 42.7).

Next I discuss Möller’s Algorithm (Section 42.8) and Rouillier’s Rational Universal Representation (Section 42.9), postponing the discussion of the effective algorithms computing with arihmetical expressions of roots given via algemaine and RUR representation to Chapter 45.

42.1 Ritt: Characteristic sets for differential polynomial ideals

Let k be a differential ^{1} field of characteristic zero.

Once an indeterminate, such as Y is introduced, it is implicitly considered as the first element of an infinite sequence of symbols Y, Y′, Y ”, ⋯, Y^{(p)}, ⋯; Y is then a differential indeterminate whose pth derivative is Y^{(p)}^{2}.

Once we consider n differential indeterminates Y_{1}, …, Y_{n} we will denote Y_{ij} the jth derivative of Y_{i}. We will denote

k{Y_{1}, …, Y_{n}} := k[Y_{ij} : 1 ≤ i ≤ n, j ∈ ℕ]

the polynomial ring in the infinite set of variables {Y_{ij} : 1 ≤ i ≤ n, j ∈ ℕ}, whose elements we call differential polynomials. For each A ∈ k{Y_{1}, …, Y_{n}} its derivative is the differential polynomial obtained applying the rules

(1) (a + b)′ = a′ + b′,

1 Id est a field which is endowed of an operation (differentiation) ·′ : k → k which

satisfies, for each a, b ∈ k,

(1) (a + b)′ = a′ + b′,

(2) (ab)′ = a′b + ab′. The elements a ∈ k for which a′ = 0 are called constants. Note that, setting

• b := 0 in (1) we have 0′ = 0, and

• b := 1, a ≠ 0 in (2) we have 1′ = 0.

It is then easy to deduce that from the equalities

• (m + 1)′ = m′ + 1′ = m′,

• 0 = 0′ = (m + (−m))′ = m′ + (−m)′ ⟹ (−m)′ = −(m′),

• 0 = (m · m^{−1})′ = m(m^{−1})′ + m^{−1}m′ ⟹ (m^{−1})′ = − \frac{m′}{m^{2}} , satisfied by each m ≠ 0, that each a ∈ ℚ ⊂ k is a constant.

2 And the p^{th} derivative of Y^{(q)} is Y^{(q+p)} for each integers p and q.

(2) (ab)′ = a′b + ab′,

(3) (Y_{i}^{(q)})′ := Y_{i}^{(q+1)} for each i and q.

Example 42.1.1. If k is the constant field ℚ(X) and A := XY_{1}^{2} + X^{2}Y_{21} ∈ k{Y_{1}, Y_{2}} then A′ := Y_{1}^{2} + 2XY_{1}Y_{11} + 2XY_{21} + X^{2}Y_{22}, and

A” := 4Y_{1}Y_{11} + 2XY_{11}^{2} + 2XY_{1}Y_{12} + 2Y_{21} + 4XY_{22} + X^{2}Y_{23}.

�

Definition 42.1.2. A subset I ⊂ k{Y_{1}, …, Y_{n}} is called a differential ideal if it is an ideal and satisfies

f ∈ I ⟹ f′ ∈ I.

Remark 42.1.3. Note that given any set Λ ⊂ k{Y_{1}, …, Y_{n}} the (polynomial) ideal generated by Λ does not necessarily coincide with the differential ideal which can be only defined as, equivalently,

(1) the set I ⊂ k{Y_{1}, …, Y_{n}} such that • Λ ⊂ I

• G_{1}, G_{2} ∈ I ⟹ G_{1} + G_{2} ∈ I

• G ∈ I, A ∈ k{Y_{1}, …, Y_{n}} ⟹ AG ∈ I

• G ∈ I ⟹ G′ ∈ I;

(2) the smallest differential ideal containing Λ;

(3) the intersection of all differential ideals containing Λ.

�

Historical Remark 42.1.4. In connection with Historical Remark 30.2.6 it is worthwhile to compare Ritt’s notation which, for a set Λ ⊂ k{Y_{1}, …, Y_{n}}, denotes

(Λ) the polynomial ideal generated by it; [Λ] the differential ideal generated by it^{3}; {Λ} the radical of [Λ] which is in fact a differential ideal^{4}.

3 which in fact (compare (3) in the remark above) is an intersection of differential

ideals. 4 The argument consists in proving that, for each π ∈ ℕ, π ≠ 0, and each differen-

tial polynomial A ∈ k{Y_{1}, …, Y_{n}} the following holds:

(1) A^{π−1}A′ ∈ [A^{π}];

(2) A^{π−δ}A^{′2δ−1} ∈ [A^{π}] ⟹ A^{π−δ−1}A^{′2δ+1} ∈ [A^{π}], for each δ ∈ ℕ, 1 ≤ δ < π,

(3) A^{′2π−1} ∈ [A^{π}],

(4) A ∈ {Λ} ⟹ A′ ∈ {Λ}.

In fact:

(1) A^{π−1}A′ = π^{−1}(A^{π})′ ∈ [A^{π}];

(2) B := (π − δ)A^{π−δ−1}A^{′2δ} + (2δ − 1)A^{π−δ}A^{′2δ−2}A” = (A^{π−δ}A^{′2δ−1})′ ∈ [A^{π}] so that,

Personally, I think that this notation is not a remainder of Steinitz’ but

an elementary direct use of the obvious sequence (·), [·], {·}.  �

Definition 42.1.5 (Ritt). For a polynomial A ∈ k{Y_{1}, …, Y_{n}}

• the class of A, class(A), is the value p ≤ n such that

A ∈ k{Y_{1}, …, Y_{p}} \ k{Y_{1}, …, Y_{p−1}};

• the order of A w.r.t. Y_{i} is the value j such that^{5}

A ∈ F[Y_{i}, Y_{i1}, …, Y_{ij}] \ F[Y_{i}, Y_{i1}, …, Y_{i j−1}]

where we set F := k{Y_{1}, …, Y_{i−1}, Y_{i+1}, …, Y_{n}}.

If A ∈ k, A is said to be of class 0.

If A ∈ k{Y_{1}, …, Y_{i−1}, Y_{i+1}, …, Y_{n}}, its order w.r.t. Y_{i} is 0.

For A_{1}, A_{2} ∈ k{Y_{1}, …, Y_{n}}, A_{1} is said to be of higher rank than A_{2} in Y_{i} if either A_{1} is of higher order than A_{2} w.r.t. Y_{i} or A_{1} and A_{2} have the same order δ but the degree of A_{1} in the variable Y_{iδ} is higher than that of A_{2}.

If class(A_{1}) = p > 0, A_{2} will be said reduced w.r.t. A_{1} if it is of lower

rank in Y_{p} than A_{1}.  �

Definition 42.1.6 (Ritt). Let A_{1}, A_{2} ∈ k{Y_{1}, …, Y_{n}} and denote, for i ∈ {1, 2}

p_{i} := class(A_{i}) the class of A_{i}, δ_{i} the order of A_{i} w.r.t. Y_{p_{i}}, d_{i} the degree of A_{i} in the variable Y_{p_{i}δ_{i}}.

A_{1} is said to be of higher rank than of A_{2} (denoteds as: A_{1} ≻ A_{2}) if

\{ \begin{matrix}p_{1} > p_{2} & or \\ p_{1} = p_{2}, δ_{1} > δ_{2} & or \\ p_{1} = p_{2}, δ_{1} = δ_{2} & and d_{1} > d_{2}. \end{matrix}

A_{1} and A_{2} are said to be of the same rank (denoted as: A_{1} ∼ A_{2}) if

p_{1} = p_{2}, δ_{1} = δ_{2} and d_{1} = d_{2}.  �
(π − δ)A^{π−δ−1}A^{′2δ+1} = A′B − (2δ − 1)A^{π−δ}A^{′2δ−1}A” ∈ [A^{π}].

(3) Since A^{π−δ}A^{′2δ−1} ∈ [A^{π}] for δ = 1 by (1) iteratively (2) implies

A^{π−δ−1}A^{′2δ+1} ∈ [A^{π}]

for δ = π − 1 id est A^{′2π−1} ∈ [A^{π}].

(4) Let π be such that A^{π} ∈ [Λ]; then by the previous result A^{′2π−1} ∈ [A^{π}] ⊂ [Λ] whence A′ ∈ {Λ}.

5 We must consider also the case j = 0, where, with a slight abuse of notation, we set Y_{i0} := Y_{i}.

Both ⪯ and ∼ are equivalences.

Remark also that the only termordering which is compatible with ≺ is the lexicographical order < induced by

Y_{1} < Y_{12} < ⋯ < Y_{1j} < ⋯ < Y_{2} < Y_{22} < ⋯ < Y_{n} < Y_{n2} < ⋯

Lemma 42.1.7. Each set of differential polynomials A contains a member A ∈ A such that A ⪯ B for each B ∈ A.

Proof. If A ∩ k ≠ ∅ any element there answers the requirement. Otherwise denote

• p the minimal value such that A ∩ k{Y_{1}, …, Y_{p}} ≠ ∅,

• δ the minimal value such that B := A ∩ k{Y_{1}, …, Y_{p−1}}[Y_{p}, Y_{p1}, …, Y_{pδ}] ≠ ∅

and choose in B the element of minimal degree in Y_{pδ}.  �

Definition 42.1.8 (Ritt). A finite set {A_{1}, …, A_{r}} of differenatial polynomials is called a chain if either

• r = 1 and A_{1} ≠ 0 or

• r > 1, class(A_{1}) = p > 0, and, for each j > i, class(A_{j}) > class(A_{i}) and A_{j} is reduced w.r.t. A_{i}^{6}.

The chain A := {A_{1}, …, A_{r}} is said to be of highest rank than the chain B := {B_{1}, …, B_{s}} (denoted A ≻ B) if either

(1) there is j, j ≤ min{r, s} such that A_{i} ∼ B_{i} for i < j and A_{j} ≻ B_{j}, or (2) s > r, and A_{i} ∼ B_{i} for i ≤ r

The chains A := {A_{1}, …, A_{r}} and B := {B_{1}, …, B_{s}} are said to be of the same rank (denoted A ∼ B) iff s = r, and A_{i} ∼ B_{i} for each i.

If A := {A_{1}, …, A_{r}} is a chain for which class(A_{1}) = p > 0, a differetial polynomial F will be said reduced w.r.t. A if it is reduced w.r.t. each A_{i} ∈ A.

�

Lemma 42.1.9. Let

A := {A_{1}, …, A_{r}}, B := {B_{1}, …, B_{s}}, C := {C_{1}, …, C_{t}}

be three chains. Then

A ≻ B, B ≻ C ⟹ A ≻ C.

Proof. There are four cases:

• Both A ≻ B and B ≻ C for the reason (1): denote j the smallst value such that B_{j} ≻ C_{j}. Either

– A_{i} ∼ B_{i} ∼ C_{i} for i < j and A_{j} ⪰ B_{j} ≻ C_{j}; or

6 Of course r ≤ n.

– there is h ≤ j such that A_{i} ∼ B_{i} ∼ C_{i} for i < h and A_{h} ≻ B_{h} ⪰ C_{h}. In both cases A ≻ C by (1).

• A ≻ B by (2), while B ≻ C by (1), and let j the smallest value such that B_{j} ≻ C_{j}:

– If r < j ≤ t, then A ≻ C by (2);

– If r ≥ j, then A_{i} ∼ B_{i} ∼ C_{i} for i < j and A_{j} ∼ B_{j} ≻ C_{j}, so that A ≻ C

by (1);

• A ≻ B by (1), while B ≻ C by (2) and let j the smallest value such that A_{j} ≻ B_{j}: then A_{i} ∼ B_{i} ∼ C_{i} for i < j and A_{j} ≻ B_{j} ∼ C_{j}; therefore A ≻ C by (1);

• both A ≻ B and B ≻ C by (2) so that r < s < t, A_{i} ∼ B_{i} ∼ C_{i} for i ≤ r

and A ≻ C by (2).  �

Lemma 42.1.10. Each set of chains 𝔄 contains a member A ∈ 𝔄 such that A ⪯ B for each B ∈ 𝔄.

Proof. We form a subset 𝔄_{1} ⊂ 𝔄 putting in 𝔄_{1} the chains A := {A_{1}, …, A_{r}} which satisfy A_{1} ⪯ B_{1} for each B := {B_{1}, …, B_{s}} ∈ 𝔄.

If each chain A ∈ 𝔄_{1} satisfies #A = 1 any chain in 𝔄_{1} satisfies our requirement.

Otherwise, we form a subset 𝔄_{2} ⊂ 𝔄_{1} collecting the chains A := {A_{1}, …, A_{r}} which satisfy A_{2} ⪯ B_{2} for each B := {B_{1}, …, B_{s}} ∈ 𝔄_{1}. If each chain A ∈ 𝔄_{2} satisfies #A = 2 any chain in 𝔄_{2} serves our purpose.

Otherwise we repeat the same construction; since each chain has at most

n elements, in the worst case 𝔄_{n} returns the required chains.  �

Definition 42.1.11. For any (finite or infinite) set G ⊂ k{Y_{1}, …, Y_{n}}, any chain A ⊂ G such that A ⪯ B for each chain B ⊂ G, whose existence is proved in the Lemma above, is called a characteristic set of G.

Note that (A) ⊆ (G) but equality does not necessarily hold.

Lemma 42.1.12. Let G ⊂ k{Y_{1}, …, Y_{n}} and A := {A_{1}, …, A_{r}} ⊂ G be a chain, where class(A_{1}) > 0. The following conditions are equivalent:

(1) A is a characteristic set of G,

(2) G contains no G ∈ k{Y_{1}, …, Y_{n}} \ {0} which is reduced w.r.t. A.

Proof. Assume that A is not a characteristic set and let B := {B_{1}, …, B_{s}} be a characteristic set, so that B ≺ A. If A ≻ B by (1), there is some B_{i}, i ≤ r, such that B_{i} ≺ A_{i} so that is reduced by A; if, instead, A ≻ B by (2), B_{r+1} is reduced by A.

Suppose now that G contains a differential polynomial G ≠ 0 which is reduced w.r.t. A. If class(G) > class(A_{r}), then the chain B := {A_{1}, …, A_{r}, G} is lower than A; otherwise, denoting j be the highest value for which the class(A_{j}) ≤ class(G), the chain B := {A_{1}, …, A_{j}, G, A_{j+1}, …, A_{r}} is lower

than A.  �

Lemma 42.1.13. Let G ⊂ k{Y_{1}, …, Y_{n}} and A := {A_{1}, …, A_{r}} ⊂ G be a characteristic set of G, where class(A_{1}) > 0. Let G ∉ G be a nonzero differential polynomial which is reduced w.r.t. A, G′ := G ∪ {G} and B be a

characteristic set of G′. Then B ≺ A.  �

Algorithm 42.1.14. Let G ⊂ k{Y_{1}, …, Y_{n}} \ {0} be a finite set. The following algorithm allows to extract a characteristic set A := {A_{1}, …, A_{r}} ⊂ G.

Let us begin by picking an element A_{1} ∈ G which is of least rank. If class(A_{1}) = 0, A := {A_{1}} is the required characteristic set. If class(A_{1}) > 0 and no element in G is reduced w.r.t. {A_{1}}, again A := {A_{1}} is the required characteristic set.

Otherwise, each element in G which is reduced w.r.t. {A_{1}} is such that class(G) > class(A_{1}); choose as A_{2} any such element of less rank.

Again, either G contains no other element which is reduced w.r.t. A := {A_{1}, A_{2}} which is the required characteristic set; or one can choose as A_{3} any element which is reduced w.r.t. A := {A_{1}, A_{2}} and of minimal rank among all possible choices.

Inductively repeating the same constructions, a characeristic set A := {A_{1}, …, A_{r}} is obtained in a finite number of steps.

�

Definition 42.1.15. For a differential polynomial G ∈ k{Y_{1}, …, Y_{n}} of class p > 0 and of order m in Y_{p} the separant of G is the differential polynomial\frac{∂G}{∂Y_{pm}} and its initial the cofficient of the highest power of Y_{pm} in G.

More precisely, expressing G as a univariate polynomial

G := ∑_{i=0}^{d} c_{i}Y_{pm}^{i} ∈ k{Y_{1}, …, Y_{p−1}}[Y_{p}, Y_{p1}, …, Y_{p m−1}][Y_{pm}],

with c_{d} ≠ 0, its separant is \frac{∂G}{∂Y_{pm}} = ∑_{i=1}^{d} ic_{i}Y_{pm}^{i−1} and its initial is c_{d}.  �

Let A := {A_{1}, …, A_{r}} be a chain and let us denote, for each i, S_{i} and I_{i} the separant and initial of A_{i}.

If a differential polynomial G is not reduced w.r.t. A, let us denote

j the greatest value such that G is not reduced w.r.t. A_{j}, p the class of A_{j}, m the order of A_{j} in Y_{p}; h ≥ m the order of G in Y_{p};

we can therefore associate to each differential polynomial G, which is not reduced w.r.t. A, a couple Φ(G) := (j, h), 1 ≤ j ≤ r, h ∈ ℕ and assume that the set of such couples is well-ordered by the ordering < defined by

(j, h) > (j′, h′) ⟺ either j > j′ or j = j′ and h > h′.

Let us consider the possible cases:

(1) If h > m, denote l := h − m and remark that A_{j}^{(l)} is of order h in Y_{p}, linear in Y_{ph} and with S_{j} as initial. The division algorithm performed in

k{Y_{1}, …, Y_{p−1}}[Y_{p}, Y_{p1}, …, Y_{p m−1}][Y_{pm}]

(Compare vol. 1, page 12) allows to compute a value v ∈ ℕ and differential polynomials C, D satisfying

S_{j}^{v}G = CA_{j}^{(l)} + D;

remark that D is

(a) uniquely determined, if v is chosen as small as possible,

(b) of order less than h in Y_{p},

(c) of rank not higher then G in Y_{a}, p < a ≤ n^{7},

(d) and so reduced w.r.t. A_{i}, i > j. Thus Φ(D) := (j′, h′) < (j, h) = Φ(G) since, either h′ < h or h′ = m, and D is reduced by A_{i} for each i > j so that j′ ≤ j

(2) If h = m, then both G and A_{j} can be considered as univariate polynomials in

k{Y_{1}, …, Y_{p−1}, Y_{p+1}, …, Y_{n}}[Y_{p}, Y_{p1}, …, Y_{p m−1}][Y_{pm}],

where the division algorithm allows to compute a value v ∈ ℕ and differential polynomials C, D satisfying

I_{j}^{v}G = CA_{j} + D;

remark that D is

• uniquely determined, if v is chosen as small as possible,

• reduced w.r.t. both A_{j}

• and each A_{i}, i > j. Thus Φ(D) := (j′, h′) < (j, h) = Φ(G) since j′ < j.

Since < is noetherin, in a finite number of applications of this algorithm we can compute a sequence of differential polynomials G := D_{0}, D_{1}, …, D_{t} where D_{t} is reduced w.r.t. A and which satisfy relations

P_{i}^{v_{i}}D_{i} = C_{i}B_{i} + D_{i+1},

with

v_{i} ∈ ℕ,

7 In fact, since S_{j} is free of Y_{a}, we need only to treat the case in which G depends on Y_{a} and its derivates; denoting g the order of G in Y_{a}, clearly the order of D in Y_{a} does not exceed g and, in case its value is exactly g, the assumption that his degree δ in Y_{ag} is higher of the one of G implies that some term of C is divisible by Y_{ag}^{δ} so that some term of CA_{j}^{(l)} is divisible by Y_{ph}Y_{pg}^{δ}; this is a contradiction since no such term occurs neither in S_{j}^{v}G — G has no term divisible by Y_{pg}^{δ} — nor in D which has no term divisible by Y_{ph}.

P_{i} ∈ {S_{j}, I_{j}, 1 ≤ j ≤ r}, C_{i} ∈ k{Y_{1}, …, Y_{n}}, B_{i} ∈ {A_{j}^{(h)}, 1 ≤ j ≤ r, h ∈ ℕ}.

As a consequence

Theorem 42.1.16 (Ritt). Let A := {A_{1}, …, A_{r}} be a chain and let us denote, for each i, S_{i} and I_{i} the separant and initial of A_{i}. For each differential polynomial G it is possible to compute

values v_{i} ∈ ℕ, and w_{i} ∈ ℕ, polynomials C_{jh} ∈ k{Y_{1}, …, Y_{n}}, a polynomial R ∈ k{Y_{1}, …, Y_{n}},

such that

(1) R is reduced w.r.t. A,

(2) and is uniquely determined by the values v_{i} and w_{i}

(3) the set {(j, h) : C_{jh} ≠ 0} is finite,

(4) S_{1}^{v_{1}} ⋯ S_{r}^{v_{r}}I_{1}^{w_{1}} ⋯ I_{r}^{w_{r}}G = R + ∑_{j,h} C_{jh}A_{j}^{(h)}.

Moreover, the values v_{j} and w_{j} can be assumed to be the minimal values

satisfying a relation of this kind.  �

Definition 42.1.17 (Ritt). The unique polynomial R determined by the minimal values v_{i} and w_{i} satisfying Theorem 42.1.16.(4) is called the remainder of G w.r.t. A.

Theorem 42.1.18 (Ritt). Let

I ⊂ k{Y_{1}, …, Y_{n}} be a differential ideal, A := {A_{1}, …, A_{r}} be a characteristic set of I, S_{i} and I_{i}, 1 ≤ i ≤ r, the separant and initial of A_{i}, X := ∏_{i=1}^{r} S_{i}I_{i}.

For each G ∈ k{Y_{1}, …, Y_{n}}, denoting R the remainder of G w.r.t. A, we have

G ∈ I ⟹ R = 0 ⟹ G ∈ I : X^{∞}.

If, moreover I is prime, then G ∈ I ⟺ R = 0 and (A) = I.

Proof. If G ∈ I then R ∈ I; therefore, being reduced w.r.t. A, it is necessarily 0 by Lemma 42.1.12.

Conversely if R = 0 then, with the notation of Theorem 42.1.16 we have

S_{1}^{v_{1}} ⋯ S_{r}^{v_{r}}I_{1}^{w_{1}} ⋯ I_{r}^{w_{r}}G = R + ∑_{j,h} C_{jh}A_{j}^{(h)} ∈ I

and G ∈ I : X^{∞}.

Remark now that the S_{i}s and I_{i}s being reduced w.r.t. A are not members of I (Lemma 42.1.12). Therefore, if I is prime and R = 0,

S_{1}^{v_{1}} ⋯ S_{r}^{v_{r}}I_{1}^{w_{1}} ⋯ I_{r}^{w_{r}}G = R + ∑_{j,h} C_{jh}A_{j}^{(h)} ∈ I ⟹ G ∈ I.

�

Remark 42.1.19. For any prime differential ideal I ⊂ k{Y_{1}, …, Y_{n}}, one can pick a maximal set of variables {V_{1}, …, V_{d}} ⊂ {Y_{1}, …, Y_{n}} such that

• I ∩ k{V_{1}, …, V_{d}} = 0,

• for each Z ∈ {Y_{1}, …, Y_{n}} \ {V_{1}, …, V_{d}} there is a nonzero differential polynomial G_{Z} ∈ I ∩ k{V_{1}, …, V_{d}, Z}.

Up to a relabeling the variables, we have an identification

k{Y_{1}, …, Y_{n}} ≅ k{V_{1}, …, V_{d}, Z_{1}, …, Z_{r}}

so that

• I ∩ k{V_{1}, …, V_{d}} = 0,

• for each i, 1 ≤ i ≤ r, there is a nonzero differential polynomial in I ∩ k{V_{1}, …, V_{d}, Z_{i}}.

If for each i, 1 ≤ i ≤ r, we pick any element A_{i} ∈ I ∩ k{V_{1}, …, V_{d}, Z_{i}} then the set {A_{1}, …, A_{r}} is naturally a chain, since each A_{i} is reduced w.r.t. {A_{1}, …, A_{i−1}}.

Analogously, if we pick any element A_{1} ∈ I ∩ k{V_{1}, …, V_{d}, Z_{1}} and, recursively, for i, 1 < i ≤ r, any element A_{i} ∈ I ∩ k{V_{1}, …, V_{d}, Z_{1}, …, Z_{i}} which is reduced w.r.t. {A_{1}, …, A_{i−1}} and of least rank among all possible choices^{8}, then {A_{1}, …, A_{r}} is a characteristic set of I.

We will call {V_{1}, …, V_{d}} — following Weispfening (cf. Definition 27.11.1)

— a maximal set of independent indeterminates or — following Ritt — a parametric set of indeterminates or — following Lazard — the set of the

trascendental variables for I.  �

8 This means that we must pick a reduced polynomial

A_{i} ∈ k{V_{1}, …, V_{d}, Z_{1}, …, Z_{i−1}}[Z_{i}, Z_{i1}, ⋯, Z_{i q−1}][Z_{iq}]

of minimal degree in Z_{iq} among all possible choices, where q, the order of A_{i} in Z_{i}, is the minimal value for which

I ∩ k{V_{1}, …, V_{d}, Z_{1}, …, Z_{i−1}}[Z_{i}, Z_{i1}, ⋯, Z_{i q−1}][Z_{iq}] ≠ ∅.

42.2 Ritt: Characteristic sets for polynomial ideals

Let us now restrict ourselve to an (algebraic) field k of characteristic zero, without requiring the existence of a differential structure and the polynomial ring P := k[X_{1}, …, X_{n}] and reinterpret the theory developped in the previous Section, assuming that k is a differential field in which all derivatives are zero and P as a subring of k{X_{1}, …, X_{n}}:

k[X_{1}, …, X_{n}] ⊂ k{X_{1}, …, X_{n}}.

Then for any set Λ ⊂ k[X_{1}, …, X_{n}] we restrict ourselves to consider the polynomial ideal

𝕀(Λ) := [Λ] ∩ k[X_{1}, …, X_{n}] = (Λ) ∩ k[X_{1}, …, X_{n}].

In this context the same notation and results introduced in the previous Section are still availale. In particular:

• for a polynomial A ∈ k[X_{1}, …, X_{n}] the class of A, class(A), is the value p ≤ n such that

A ∈ k[X_{1}, …, X_{p}] \ k[X_{1}, …, X_{p−1}],

the polynomials of class 0 being the elements in k;

• for any two polynomials A_{1}, A_{2} ∈ k[X_{1}, …, X_{n}], A_{1} is said to be of higher rank than A_{2} in X_{i} if the degree of A_{1} in the variable X_{i} is higher than that of A_{2};

• if class(A_{1}) = p > 0, A_{2} will be said reduced w.r.t. A_{1} if it is of lower degree in X_{p} than A_{1};

• for any two polynomials A_{1}, A_{2} ∈ k[X_{1}, …, X_{n}], denoting, for i ∈ {1, 2}, p_{i} the class of A_{i} and d_{i} the degree of A_{i} in the variable X_{p_{i}},

A_{1} ≻ A_{2} ⟺ \{ \begin{matrix}p_{1} > p_{2} & or \\ p_{1} = p_{2} & and d_{1} > d_{2} \end{matrix}

and

A_{1} ∼ A_{2} ⟺ p_{1} = p_{2}, d_{1} = d_{2};

• a finite set

A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}]

is called a chain (or ascending set, or reduced triangular set) if either r = 1 and A_{1} ≠ 0 or

– class(A_{1}) = p > 0,cl

– for each j > i, class(A_{j}) > class(A_{i}), and

– A_{j} is reduced w.r.t A_{i}; • given any (finite or infinite) set G ⊂ k[X_{1}, …, X_{n}], in the chain

A := {A_{1}, …, A_{r}} ⊂ G

produced by Algorithm 42.1.14, each element A_{i} is not only reduced w.r.t. {A_{1}, ⋯, A_{i−1}} and of minimal rank among all possible choices, but necessarily his class is higher of the one of A_{i−1};

• for two chains in k[X_{1}, …, X_{n}], we have

{A_{1}, …, A_{r}} := A ≻ B := {B_{1}, …, B_{s}}

if either

(1) there is j, j ≤ min{r, s} such that A_{i} ∼ B_{i} for i < j and A_{j} ≻ B_{j}, or (2) s > r, and A_{i} ∼ B_{i} for i ≤ r and

A ∼ B ⟺ s = r, and A_{i} ∼ B_{i} for each i;

• if A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}] is a chain for which class(A_{1}) = p > 0, a polynomial F ∈ k[X_{1}, …, X_{n}] is reduced w.r.t. A if it is reduced w.r.t. each A_{i} ∈ A;

• for any set G ⊂ k[X_{1}, …, X_{n}], any chain A ⊂ G such that A ⪯ B for each chain B ⊂ G is called a characteristic set (or: basic set) of G;

• for a polynomial

G := ∑_{i=0}^{d} c_{i}X_{p}^{i} ∈ k[X_{1}, …, X_{p−1}][X_{p}], c_{i} ∈ k[X_{1}, …, X_{p−1}], c_{d} ≠ 0

of class p > 0 its initial is its leading polynomial c_{d} = Lp(G).

• For a chain A := {A_{1}, …, A_{r}} and a polynomial G not reduced w.r.t. A, denoting, for each i, I_{i} the initial of A_{i}, let j the greatest value such that G is not reduced w.r.t. A_{j}, and p the class of A_{j}, then the division algorithm in

k[X_{1}, …, X_{p−1}, X_{p+1}, …, X_{n}][X_{p}],

allows to compute a value v ∈ ℕ and polynomials C, D satisfying

I_{j}^{v}G = CA_{j} + R,

where R is

– uniquely determined, if v is chosen as small as possible,

– reduced w.r.t. both A_{j}

– and each A_{i}, i > j.

• Then (compare Theorem 42.1.16) for each such G it is possible to compute values w_{i} ∈ ℕ, polynomials C_{j} ∈ k[X_{1}, …, X_{n}], a polynomial R ∈ k[X_{1}, …, X_{n}], such that

(1) R is reduced w.r.t. A, (2) and is uniquely determined by the values w_{i}

(3) I_{1}^{w_{1}} ⋯ I_{r}^{w_{r}}G = R + ∑_{j} C_{j}A_{j}. Moreover, the values v_{i} can be assumed to be the minimal values satisfying a relation of this kind.

• The remainder of G w.r.t. A is the unique polynomial R determined by the minimal values v_{i} in the formula above.

• For any prime ideal I ⊂ k[X_{1}, …, X_{n}], we can relabel the variables so that

k[X_{1}, …, X_{n}] ≅ k[V_{1}, …, V_{d}, Z_{1}, …, Z_{r}]

and {V_{1}, …, V_{d}} is a maximal set of independent indeterminates for I, d := dim(I). Then each characteristic set of I consists (Remark 42.1.19) of r polynomials A_{i} ∈ I ∩ k[V_{1}, …, V_{d}, Z_{1}, …, Z_{i}] which are reduced w.r.t. {A_{1}, …, A_{i−1}} and of minimal degree in Z_{i} among all possible choices.

In this context it is worthwhile to record an old-fashioned proof of the following well-known result:

Proposition 42.2.1. Let I ⊂ k[X_{1}, …, X_{n}] be a prime ideal, dim(I) := d, {V_{1}, …, V_{d}} be a maximal set of independent indeterminates] for I and K ∈ k[X_{1}, …, X_{n}] a polynomial not contained in I.

Then the ideal I′ := I + 𝕀(K) is such that I′ ∩ k[V_{1}, …, V_{d}] ≠ {0}.

Proof (Ritt). Using the same notation as above, this is Ritt’s argument^{9}:

We start with the observation that the polynomials in I which involve no Z_{i} with i > j, where 1 ≤ j < n−d, constitute a prime polynomial ideal; we describe this prime ideal^{10} by I_{j}. I′ contains the remainder of K with respect to {A_{1}, …, A_{r}}. Of all nonzero polynomials in I′ which are reduced with respect to {A_{1}, …, A_{r}}, let B the one which is of lowest rank. We say that B is free of the Z. Suppose that this is not so, and let B of class d + p with p > 0. The initial C of B is not in I. There is a relation

C^{m}A_{p} = DB + E

where E, if not zero, is of lower degree than B in Z_{p}. We say that E is in I. Let this be false. If p > 1, the remainder of E with respect to A_{1}, …, A_{p−1} is a non zero polynomial contained in I′, which is reduced with respect to {A_{1}, …, A_{r}} and of lower rank than B. If p = 1, a similar statement can be made of E itself. Thus^{11} E is in

9 Ritt J.F., Differential Algebra, A.M.S. Colloquium Publications 33 (1950) p.84.

I just adapted the notation. 10 which is I_{j} := I ∩ k[V_{1}, …, V_{d}, Z_{1}, …, Z_{j}]. 11 Remark that, by assumption, B is a nonzero polynomial contained in I′, which is

reduced with respect to {A_{1}, …, A_{r}} and of lower rank. So we have just reached a contradiction.

I, so that DB is in I. Then^{12} D is in I. D is of positive degree^{13} in Z_{p}. As the initial of DB is that of C^{m}A_{p}, the initial I of D is not in I. If we had p = 1, D would be a nonzero polynomial in I which is reduced with respect to {A_{1}, …, A_{r}}; this is because D is of lowest degree in X_{p} than A_{p}. Thus p > 1. The remainder of D with respect to A_{1}, …, A_{p−1} is zero^{14}. Thus JD, with J some product of powers of the initials of A_{1}, …, A_{p−1}, is linear^{15} in A_{1}, …, A_{p−1}. If we write JD as a polynomial in Z_{p} its coefficients will be in I_{p−1}. Thus JI is in I_{p−1}. This is false because neither J nor I is in I_{p−1}.

Thus B is free of the Z and our statement is proved.  �

Remark 42.2.2. Since any polynomial P = P_{0} can be uniquely expressed as

P = P_{0} = Lp(P_{0})X_{j_{0}}^{δ_{0}} + R_{0}

where j_{0} = class(P_{0}), P_{1} := Lp(P) ∈ k[X_{1}, …, X_{j_{0}−1}] and deg_{j_{0}}(R_{0}) < δ_{0} = deg_{j_{0}}(P_{0}), recursively, we can define

• values j_{i} := class(P_{i}) < j_{i−1}, and δ_{i} ∈ ℕ,

• polynomials R_{i} ∈ k[X_{1}, …, X_{j_{i}}] and

• P_{i+1} := Lp_{i}(P) := Lp(Lp_{i−1}(P) = Lp(P_{i}) ∈ k[X_{1}, …, X_{j_{i}−1}] such that

P_{i} = Lp_{i}(P)X_{j_{i}}^{δ_{i}} + R_{i}, deg_{j_{i}}(R_{i}) < δ_{i} = deg_{j_{i}}(P_{i})

until i reachs a value ι for which P_{ι+1} = Lp_{ι}(P) ∈ k.

Then, we have

P = Lp_{ι}(P)X_{j_{ι}}^{δ_{ι}}X_{j_{ι−1}}^{δ_{ι−1}} ⋯ X_{j_{1}}^{δ_{1}}X_{j_{0}}^{δ_{0}} + R_{ι} ∏_{i=0}^{ι−1} X_{j_{i}}^{δ_{i}} + ∑_{h=0}^{ι−1} R_{h} ∏_{i=0}^{h−1} X_{j_{i}}^{δ_{i}}.

Recalling that the only termordering which is compatible with ≺ is the lexicographical order < induced by X_{1} < X_{2} < ⋯ < X_{n}, we necessarily have

lc_{<}(P) = Lp_{ι}(P), T_{<}(P) = X_{j_{ι}}^{δ_{ι}}X_{j_{ι−1}}^{δ_{ι−1}} ⋯ X_{j_{1}}^{δ_{1}}X_{j_{0}}^{δ_{0}}.

�

If we consider a chain

A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}] ≅ k[V_{1}, …, V_{d}, Z_{1}, …, Z_{r}],

each A_{i} being of class d+i, we can find conditions for A to be a characteristic set of a prime ideal.

Let us denote

12 B ∈ k[V_{1}, …, V_{d}] ⟹ B ∉ I. 13 Necessarily the degree in Z_{p} of B is lower of that of A_{p}. 14 D ∈ I. 15 Id est JD ∈ (A_{1}, …, A_{p−1}) = 𝕀(A_{1}, …, A_{p−1}).

I the ideal generated by A in k(V_{1}, …, V_{d})[Z_{1}, …, Z_{r}], I_{j} := I ∩ k(V_{1}, …, V_{d})[Z_{1}, …, Z_{j}], L_{j} := k(V_{1}, …, V_{d})[Z_{1}, …, Z_{j}]/I_{j}, π_{j} : k(V_{1}, …, V_{d})[Z_{1}, …, Z_{r}] → L_{j}[Z_{j+1}, …, Z_{r}], I := ∏_{i} I_{i}, each I_{i} denoting the initial of A_{I}.

Then:

Proposition 42.2.3. With the present notation, A is a characteristic set of the prime ideal I : I^{∞} iff the following conditions hold:

(1) if r = 1, A_{1} is irreducible in k(V_{1}, …, V_{d})[Z_{1}].

(2) if r > 1, {A_{1}, …, A_{r−1}} is a characteristic set of the prime ideal I_{r−1} and π_{r−1}(A_{r}) is irreducible in L_{r−1}[Z_{r}].

Proof. A reformulation of Theorem 34.1.2 and Theorem 34.3.2.  �

Algorithm 42.2.4 (Ritt). Let G ⊂ k[X_{1}, …, X_{n}] be a finite set generating an ideal I.

The following algorithm allows to compute a characteristic set A := {A_{1}, …, A_{r}} of I and a polynomial I such that, denoting L the ideal generated by A we have

L ⊂ I ⊂ L : I^{∞} =: H;

if moreover I is prime, then I = L.

One begins by extracting from G a characteristic set A with the methode described in Algorithm 42.1.14, so that (A) ⊆ I^{16}. Next he computes the remainders w.r.t. A of each member in G and includes the nonzero ones in G producing a larger set G′ which however satisfies 𝕀(G′) = 𝕀(G) = I.

If not all such remainders are zero, then (Lemma 42.1.13) a characteristic set A′ of G′ satisfies A′ ≺ A.

Thus the same procedure can be repeated until giving a set G^{∗} which satisfies 𝕀(G^{∗}) = 𝕀(G) = I and a characteristic set A^{∗} := {A_{1}, …, A_{r}} extracted from (G^{∗}), for which either

• A_{1} is of class zero so that I = (1), or

• all the remainders w.r.t. A^{∗} of each member in G^{∗} is zero. Then (Theorem 42.1.18), denoting I := ∏_{i=1}^{r} I_{i} the product of all initials I_{i} of the A_{i}s, we have

𝕀(A^{∗}) ⊂ I ⊂ 𝕀(A^{∗}) : I^{∞};
if moreover I is prime, 𝕀(A^{∗}) = I.  �

16 We can of course assume that A_{1} is of positive class, otherwise I = (1) and we are through.

Historical Remark 42.2.5. To put in better historical perspective, I think it is worthwhile to quote the words of Ritt^{17} referring also to his older book^{18}:

The form in which the results of differential algebra are being presented has thus being deeply influenced by the teachings of Emmy Noether, a prime mover of our period, who, in continuing Julius König’s development of Kronecker’s ideas, brought mathematicians to know algebra as it was never known before. In this connection, I should like to say something concerning basis theorems. The basis theorem […] will be see to play, in the present theory, the role held by Hilbert’s theorem in the theories of polynomials ideals and of algebraic manifolds. When I began to work on algebraic differential equations, early in 1930, van der Waerden’s excellent Modern Algebra had not yet appeared. However, Emmy Noether’s work of the twenties was available, and there was nothing to prevent one from learning in her papers the value of basis theorems in decomposition problems. Actually, I became acquainted with the basis theorem principle in the writings of Jules Drach on logical integration^{19}.

�

42.3 Ritt’s and Wu’s Solvers

Algorithm 42.3.1 (Ritt). Ritt applied Proposition 42.2.3, Algorithm 42.2.4 and Theorem 42.1.18 as tools for proposing a solver^{20} which, given a finite set G ⊂ k[X_{1}, …, X_{n}] generating an ideal I, applyes Algorithm 42.2.4 in order to extract a characteristic set A^{∗} satisfying the properties stated by Theorem 42.1.18 and tests whether^{21} it satisfies the properties of Proposition 42.2.3.

17 In the Preface of his book Differential Algebra, A.M.S. Colloquium Publications

33 (1950), p.iv 18 Ritt J.F., Differential Equations from the Algebraic Standpoint, A.M.S. Collo-

quium Publications 14 (1932). 19 The reference is to J. Drach, Essai sur la théorie général de l’integration et sur

la classification des Trascendentes Ann. Éc. Norm. 3^{e} série 15 (1898) 245–384. 20 Ritt J.F., Differential Algebra, A.M.S. Colloquium Publications 33 (1950), p.95-

98. 21 Essentially Ritt proposes the primality test discussed in Section 35.4:

• the rôle used there by the Gröbner basis G′ is taken here by the characteristic set A^{∗};

• the test I : I^{∞} = I is not required since, in this setting

I : I^{∞} = (𝕀(A^{∗}) : I^{∞}) : I^{∞} = 𝕀(A^{∗}) : I^{∞} = I.

If the answer is positive then A^{∗} is a characteristic set of the prime ideal H := 𝕀(A^{∗}) : I^{∞} and, in this case^{22}

Z(I) = Z(H) ∪ Z(I + 𝕀(I_{1})) ∪ ⋯ ∪ Z(I + 𝕀(I_{r}));

the same algorithm is to be recursively applied to each set G ∪ {I_{i}}.

If, instead, the answer is negative, then^{23}, we have found (using the notation of Proposition 42.2.3) some factorization

π_{r−1}(A_{r}) = c ∏_{i=1}^{s} π(g_{i})^{e_{i}}

in L_{r−1} where g_{i} ∈ k[V_{1}, …, V_{d}][Z_{1}, …, Z_{r−1}][Z_{r}] and c ∈ k(V_{1}, …, V_{d}) is a unit; therefore

Z(I) = Z(I + 𝕀(g_{1})) ∪ ⋯ ∪ Z(I + 𝕀(g_{s})).

The result corresponds to an irredundant prime decomposition of I. �

Historical Remark 42.3.2. In other words, Ritt (in 1950) is applying the notion of ’solving’ discussed in Section 34.5. In 1978, Wu Wen-tsün relaxed this notion of ’solving’ preserving the structure of the corresponding basis (ascending set) which has a similar shape as a Primbasis (Definition 34.3.3) or an admissible sequence (Definition 8.2.2 and 11.4.2) but requiring less strong properties (essentially just those implied by Theorem 42.1.18 and Algorithm 42.2.4).

In fact, in his research toward a Theorem-Proving algorithm in differential (and elementary) geometry, as a tool for testing whether f(α) = 0 for each root^{24} α of I, where f ∈ k{X_{1}, …, X_{n}} is a given differential polynomial and I := 𝕀(G) ⊂ k{X_{1}, …, X_{n}} a given differential ideal, Wu applied directly Ritt’s theory reducing the problem to the test whether

(1) the remainder of f w.r.t. A^{∗} is zero and,

(2) by recursive application of the same algorithm, f(α) = 0 for each root α of I + {I_{i}}, i ≤ r.

In fact, for each root α of I,

(1) if ∏_{i=1}^{r} I_{i}(α) ≠ 0, then, by Theorem 42.1.16, f(α) = 0 iff the remainder of f w.r.t. A is zero;

(2) if I_{i}(α) = 0, then α is a root of I + 𝕀(I_{i}).

22 It is worthwhile to note that this formula which is today improperly attributed

to Wu Wen-tsün is Equation (32) in Ritt J.F., op. cit. p. 98. 23 As in the prime decomposition algorithm discussed in Section 35.2 24 If K ⊃ k is a differential field extension of k, α := (a_{1}, …, a_{n}) ∈ K^{n} and f ∈

k{X_{1}, …, X_{n}} is a differential polynomial, the evaluation f(α) is by definition the value Φ_{α}(f), where Φ_{α} : k{X_{1}, …, X_{n}} → K is the morphism defined by Φ_{α}(X_{ij}) = a_{i}^{(j)} for each i, j.

For his application, there was therefore no need of performing factorization of the characteristic set A^{∗} in order to deduce an irredundant prime decomposition of I; all one needs is to decompose \sqrt{I} as an intersection of (unmixed) ideals generated by a characteristic set, in order to apply the results implied by Theorem 42.1.18 and Algorithm 42.2.4 reducing the problem to zero-testing of ’normal forms’ of each element in G w.r.t. each such characteristic set.

In the algebraic setting k[X_{1}, …, X_{n}], such characteristic set is a set

A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}] ≅ k[V_{1}, …, V_{d}][Z_{1}, …, Z_{r}],

where (compare Definition 22.0.1), for each i,

• the degree of A_{i} in X_{j} is less than the one of A_{j}, for each j < i, and

• A_{i} ∈ k[V_{1}, …, V_{d}][Z_{1}, …, Z_{i−1}][Z_{i}]

id est it is exactly what we informally called a weak admissible sequence^{25}. �

Algorithm 42.3.3 (Wu). Within this approach, Algorithm 42.3.1 is simplified as follows: given a finite set G ⊂ k[X_{1}, …, X_{n}] generating an ideal I, one

• applyes Algorithm 42.2.4 in order to extract a basic set A^{∗} := {A_{1}, …, A_{r}};

• returns A^{∗} and the polynomial I := ∏_{i=1}^{r} I_{i} where I_{i} denotes the initial of A_{i},

• and apply the same algorithm to each set G ∪ {I_{i}}, 1 ≤ i ≤ r  �

whose rationale is based on the following

Lemma 42.3.4. Let

• A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}] a characteristic set,

• L the ideal generated by A,

• I_{j} the initial of A_{j}, for each j,

• I := ∏_{i=1}^{r} I_{i},

• Rem(A) the set^{26} of all polynomials whose remainder w.r.t. A is 0,

• Sat(A) := L : I^{∞},

• ℨ(A) := Z(A) \ Z(𝕀(I)) = {α ∈ k^{n} : A_{1}(α) = ⋯ = A_{r}(α) = 0 ≠ I(α)} .

Then Z(Sat(A)) = \overline{ℨ(A)}.  �

25 Throughout this chapter, I will preserve the language used in the previous books; so I will substitute with the neologisms of admissible Ritt/Lazard sequence the terminology introduced by Lazard and his students, which is in any case reported between parenthesis.

26 Not necessarily an ideal!

Proof. We have^{27}

Z(Sat(A)) = Z(\sqrt{(A) : I}) = \overline{Z(\sqrt{(A)} \ Z(\{I\})} = \overline{ℨ(A)}.

�

Corollary 42.3.5 (Ritt). With the same notation and assumptions as in Lemma 42.3.4 let G ⊂ k[X_{1}, …, X_{n}] be a finite set generating an ideal I such that A ⊂ I and G ⊂ Rem(A). Then:

(1) L ⊂ I ⊆ Rem(A) ⊆ Sat(A);

(2) if I is prime, then L = I = Rem(A) = Sat(A);

(3) Z(Sat(A)) = \overline{ℨ(A)};

(4) \overline{ℨ(A)} ⊆ Z(I) ⊆ Z(L);

(5) Z(I) = ℨ(A) ∪ Z(I + 𝕀(I_{1})) ∪ ⋯ ∪ Z(I + 𝕀(I_{r})).  �

Proof. (1) and (2) are just a reformulation of Theorem 42.1.18 and (3) is Lemma 42.3.4

Ad (4): \overline{ℨ(A)} = Z(Sat(A)) ⊆ Z(I) ⊆ Z(L).

Ad (5): for each α ∈ Z(I):

• ∏_{i=1}^{r} I_{i}(α) ≠ 0 ⟺ α ∈ ℨ(A);

• for some i ≤ r, we have I_{i}(α) = 0 ⟺ α ∈ Z(I + {I_{i}}).  �

Corollary 42.3.6 (Wu). With the same notation and assumptions as in Lemma 42.3.4 and Corollary 42.3.5, for any g ∈ k[X_{1}, …, X_{n}], g ∈ I if and only if

• the remainder of g w.r.t. A is 0 and

• g ∈ I + 𝕀(I_{i}) for each i, 1 ≤ i ≤ r.  �

Example 42.3.7. Let

G := {X_{1}^{6} − X_{1}^{4}, (X_{1}^{4} − 2X_{1}^{2})X_{3}, (X_{1}^{2} − 1)X_{2}X_{4} + X_{3}} ⊂ k[X_{1}, X_{2}, X_{3}, X_{4}].

27 The formula Z(I : f) ⊂ Z(I) \ Z({f}).

Z (\sqrt{(A) : I}) = \overline{Z (\sqrt{(A)}) \ Z (\{I\})} is a specialization of Z(I : f) = \overline{Z(I) \ Z(\{f\}))} — where I is a radical polynomial ideal and f a polynomial in k[X_{1}, …, X_{n}], — whose proof is the following: for g ∈ I : f and α ∈ Z(I) \ Z({f}) we have 0 = fg(α) = f(α)g(α) and f(α) ≠ 0 so that g(α) = 0. Therefore Z(I) \ Z({f}) ⊂ Z(I : f).

Conversely, assume g satisfies g(α) = 0 for each α ∈ Z(I) \ Z({f})); in other

words, for each α ∈ Z(I), f(α) ≠ 0 ⟹ g(α) = 0; thus fg(α) = f(α)g(α) = 0

for each α ∈ Z(I); as a consequence fg ∈ \sqrt{I} = I and g ∈ I : f.

Thus {g ∈ k[X_{1}, …, X_{n}] : g(α) = 0, α ∈ Z(I) \ Z({f}))} ⊂ (I : f) and

G, being a characteristic set of the ideal it generates, according Ritt’s solver (Algorithm 42.3.1) we test the irreducibility of X_{1}^{6} − X_{1}^{4} discovering the decomposition

Z(G) = Z(G + {X_{1} − 1}) ∪ Z(G + {X_{1} + 1}) ∪ Z(G + {X_{1}});

the characteristic set of Z(G+{X_{1}−1}) being {X_{1}−1, X_{3}} which is prime we have found a root (1, 0) ∈ k(X_{2}, X_{4})^{2}; in the same way, Z(G+{X_{1} +1}) gives the root (−1, 0) ∈ k(X_{2}, X_{4})^{2}; the characteristic set A := {X_{1}, X_{2}X_{4} − X_{3}} of Z(G + \{X_{1}\}) returns a root (0, \frac{X_{3}}{X_{2}} ) ∈ k(X_{2}, X_{3})^{2} of the prime (A) = (A) : X_{2}^{∞} and the decomposition

Z(G) = Z(G + {X_{1} − 1}) ∪ Z(G + {X_{1} + 1}) ∪ Z((A) : X_{2}^{∞}) ∪ Z(A + {X_{2}});

the characteristic set B := {X_{1}, X_{2}, X_{3}} of A + {X_{2}}) is prime and gives the root (0, 0, 0) ∈ k(X_{4})^{3}. So in conclusion we have found the (redundant) prime decomposition

(G) = (X_{1} − 1, X_{3}) ∩ (X_{1} + 1, X_{3}) ∩ (X_{1}, X_{2}X_{4} − X_{3}) ∩ (X_{1}, X_{2}, X_{3})

and the manifold decomposition

Z(G) = {(1, a, 0, b),: a, b ∈ k} ∪ {(−1, a, 0, b) : a, b ∈ k}
∪ \{(0, a, b, \frac{b}{a}), a, b ∈ k, a ≠ 0\} ∪ \{(0, 0, 0, b), b ∈ k\}.

Wu’s solver (Algorithm 42.3.3) instead returns the root decompositions

Z(G) = ℨ(G) ∪ Z(G + {X_{1}^{4} − 2X_{1}^{2}}) ∪ Z(G + {X_{1}^{2} − 1}),
Z(G + {X_{1}^{4} − 2X_{1}^{2}}) = ℨ(C_{1}) ∪ Z(C_{1} + {X_{2}}),
Z(C_{1} + {X_{2}}) = ℨ(C_{3}),
Z(G + {X_{1}^{2} − 1}) = ℨ(C_{2}),

where

C_{1} = {X_{1}^{2}, X_{2}X_{4} − X_{3}},
C_{2} = {X_{1}^{2} − 1, X_{3}},
C_{3} = {X_{1}^{2}, X_{2}, X_{3}},

and

\begin{aligned}ℨ(G) & = & Z(G) \ Z(\{(X_{1}^{4} − 2X_{1}^{2})(X_{1}^{2} − 1)\}) & = & ∅, \\ ℨ(C_{1}) & = & Z(C_{1}) \ Z(\{X_{2}\}) & = & \{(0, a, b, \frac{b}{a}), a, b ∈ k, a ≠ 0\}, \\ ℨ(C_{2}) & = & Z(C_{2}) & = & \{(1, a, 0, b), (−1, a, 0, b), a, b ∈ k\}. \\ ℨ(C_{3}) & = & Z(C_{3}) & = & \{(0, 0, 0, b), b ∈ k\}, \end{aligned}

�

42.4 Lazard: Triangular sets

Preserving the same notation as in Sections 42.2 and 42.3, computational considerations suggested to relax the notion of reduction as follows:

Definition 42.4.1 (Moreno Maza). For any two polynomials A_{1}, A_{2} ∈ k[X_{1}, …, X_{n}], where A_{1} is of class p > 0, A_{2} will be said initially reduced w.r.t. A_{1} if its initial is of lower degree in X_{p} than A_{1}.

Definition 42.4.2 (Aubry et al.). A finite non-empty set

{A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}]

is called

a triangular set if each A_{i} is of positive class and there are no two elements

having the same class; an initially reduced triangular set if

• A_{1} is of positive class,

• for each j > i, A_{j} is of higher class than A_{i}, and

• initially reduced w.r.t A_{i}; a fine triangular set if

• A_{1} is of positive class,

• for each j > i, A_{j} is of higher class than A_{i},

• for each j, the remainder of I_{j} w.r.t. {A_{1}, …, A_{j−1}} is not zero.

Remark 42.4.3 (Lazard). In connection with this generalization, it is easy to realize that all the results stated by Ritt for chains and characteristic sets hold verbatim for any triangular set, non dissimilarly as for Buchberger’s reduction, where the absence of interreduction of the basis does not effect the result on normal forms zero-testing.

This justifies also the relaxation of the notion of reduction.

In particular the notion of fine triangular set allows to avoid interreduction, while preserving the degree in X_{p} of an element of class p and so the

rank relation between members of triangualar sets.  �

Definition 42.4.4 (Aubry et al.). Let G ⊂ k[X_{1}, …, X_{n}] \ {0} be a finite set generating the ideal I. A finite non-empty triangular set

A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}]

is called

an admissible Ritt sequence (or: Ritt characteristic set) of G if A ≺ B for

each fine triangular set B ⊂ G; a strong admissible Ritt sequence (or: Wu characteristic set) of G if there

exists a finite set G^{∗} ⊂ I such that 𝕀(G^{∗}) = I and G^{∗} ⊆ Rem(A).

Historical Remark 42.4.5. Clearly, the notion of Ritt characteristic set is an elementary adaptation to triangular sets of the notion of characteristic set (Definition 42.1.11) and that of Wu characteristic set a precise description of the particular triangular sets which are produced by Algorithm 42.2.4; both ideas, therefore are explicitly present in Ritt’s Theory and none is related with Wu’s results.

Wu’s results are not related with the notion and properties of characteristic sets; as discussed in Historical Remark 42.3.2, his relevant contributions consist in using such theory, no more as a solving tool, but as a tool for testing membership (Corollary 42.3.6), and, in this context, to relax the irrelevant requirement of primality.

It is Lazard^{28} the person which, within the Kronecker–Duval philosophycal frame discussed in the first volume and of which he was one of the main advocates, suggested to reconsider Ritt’s solver in Wu’s relaxed context and introduced the notion of ’triangular sets’ thus strongly improving the old notion of ’solving’ as used by Kronecker, Macaualy, Gröner and Ritt and

presented in Section 34.5.  �

In this setting, let us now consider

• a triangular set A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}] generating the ideal L,

• I_{j} the initial of A_{j}, for each j,

• I := ∏_{i=1}^{r} I_{i},

• Rem(A) the set of all polynomials whose remainder w.r.t. A is 0,

• Sat(A) := L : I^{∞},

• ℨ(A) := Z(A) \ Z({I}) = {α ∈ k^{n} : A_{1}(α) = ⋯ = A_{r}(α) = 0 ≠ I(α)},

• G ⊂ k[X_{1}, …, X_{n}] be a finite set generating an ideal I such that A ⊂ I and G ⊂ Rem(A).

Corollary 42.4.6. With the present notation, if A is a strong admissible Ritt sequence of G, then

(1) L ⊂ I ⊆ Rem(A) ⊆ Sat(A);

(2) if I is prime, then L = I = Rem(A) = Sat(A);

(3) Z(Sat(A)) = \overline{ℨ(A)};

(4) \overline{ℨ(A)} ⊆ Z(I) ⊆ Z(L);

(5) Z(I) = ℨ(A) ∪ Z(I + 𝕀(I_{1})) ∪ ⋯ ∪ Z(I + 𝕀(I_{r}));

28 In Lazard D., Solving zero-dimensional algebraic systems J. Symb. Comp. 15 (1992), 117–132 Lazard D., A new method for solving algebraic systems of posisitive dimension Disc. Appl. Math. 33 (1991), 147–160 Lazard D. Systems of algebraic equations (algorithms and complexity) Symposia Mathematica 34 (1993), 84–105, Cambridge Univ. Press

(6) for each g ∈ k[X_{1}, …, X_{n}],

g ∈ I ⟺ g ∈ Rem(A) ∩ I + 𝕀(I_{1}) ∩ ⋯ ∩ I + 𝕀(I_{r}).

�

Proposition 42.4.7. If, with the present notation, A is fine, then the following conditions are equivalent

(1) A is an admissible Ritt sequence of G;

(2) I ⊂ Rem(A).

Moreover it implies

(3) A is a strong admissible Ritt sequence of G

Proof. It is just a reformulation of Lemma 42.1.12.  �

42.5 Admissible Lazard Sequence

Lazard reconsidered the notion of triangular sets in the same frame as admissible sequences (Definition 8.2.2) and admissible Duval sequences (Section 11.4).

Let us begin by explicitly interpreting the field k as a quotient field L_{0} := k of some domian R_{0}; id est we assume that we are given a domain R_{0} and a multiplicative system S_{0} ⊂ R_{0} such that^{29}

k =: L_{0} := \{\frac{a}{b} : a ∈ R_{0}, b ∈ S_{0}\}.

Then let us consider a triangular set^{30}

A := {f_{1}, …, f_{r}} ⊂ R_{0}[X_{1}, …, X_{n}]

where, by definition, we can wlog assume that

0 < class(f_{1}) < … < class(f_{i}) < class(f_{i+1}) < … < class(f_{r}).

We also set d_{i} := deg_{j}(f_{i}) where j := class(f_{i}).

We can then now partition the variables among ’parameters’ (or: ’independent indeterminates’) and the others:

29 we can for instance take R_{0} := L_{0} := k and S_{0} := {1}; but for k := ℚ we could

consider instead R_{0} := ℤ, S_{0} := ℕ \ {0}.

The reason why we choose to represent the field elements as explicit fractions with denominators in a restricted chosen multiplicative system S_{0} is in order to force uniqueness: see the note below.

30 Each element f_{i} is chosen with coefficients not in L_{0} but in R_{0}; as a consequence among all possible associated polynomials we can restrict our choice to those such that their leading coefficient is in S_{0}. If we take R_{0} := L_{0} := k and S_{0} := {1} this simply means to require that each f_{i} is monic.

Definition 42.5.1 (Lazard). A variable X_{j} is called

algebraic for A if j = class(f_{i}) for some f_{i} ∈ A which is said^{31} to introduce

X_{j};

trascendental for A if j ≠ class(f_{i}) for each f_{i} ∈ A.  �

As usual we relabel the variables as

k[X_{1}, …, X_{n}] ≅ k[V_{1}, …, V_{d}, Z_{1}, …, Z_{r}]

so that {V_{1}, …, V_{d}} (respectively {Z_{1}, …, Z_{r}}) is the set of the trascendental (respectively: algebraic) variables for A.

Then we can recursively define, for each j,

i the value such that A ∩ R_{0}[X_{1}, …, X_{j}] = {f_{1}, …, f_{i}} or, equivalently, the

maximal value for which class(f_{i}) ≤ j, δ the value such that {V_{1}, …, V_{δ}} = {X_{1}, …, X_{j}} ∩ {V_{1}, …, V_{d}}, R_{j} := R_{0}[X_{1}, …, X_{j}]/(f_{1}, …, f_{i}), S_{j} := {a ∈ R_{j} : a is not a zero-divisor}; L_{j} the quotient ring L_{j} := \{\frac{a}{b} : a ∈ R_{j}, b ∈ S_{j}\}; π_{j} := R_{0}[X_{1}, …, X_{j}] → R_{j} and π_{j} := L_{0}[X_{1}, …, X_{j}] → L_{j} the canonical

projections.

In particular, for each j

if X_{j} = Z_{i} is algebraic, so that it is introduced by f_{i} and j = class(f_{i}), we have R_{j} = R_{j−1}[X_{j}]/π_{j−1}(f_{i}) ≅ R_{0}[X_{1}, …, X_{j}]/(f_{1}, …, f_{i}), S_{j} = S_{j−1} = S_{0}[V_{1}, …, V_{δ}], L_{j} = L_{j−1}[X_{j}]/π_{j−1}(f_{i}) ≅ L_{0}(V, …, V_{δ})[Z_{1}, …, Z_{i}]/(f_{1}, …, f_{i});

if X_{j} = V_{δ} is instead trascendental, we have R_{j} = R_{0}[X_{1}, …, X_{j}]/(f_{1}, …, f_{i}) ≅ R_{j−1}[X_{j}],

S_{j} = S_{j−1}[X_{j}] = S_{0}[V_{1}, …, V_{δ}],
L_{j} = L_{j−1}(X_{j}) ≅ L_{0}(V_{1}, …, V_{δ})[Z_{1}, …, Z_{i}]/(f_{1}, …, f_{i}).

Definition 42.5.2. We say that

A := {f_{1}, …, f_{r}} ⊂ R_{0}[X_{1}, …, X_{n}]

is an admissible Lazard sequence if, for each h, 1 ≤ h ≤ r, setting j := class(f_{h}), it holds

(i) [triangular] class(f_{h}) > 0 and class(f_{h}) > class(f_{i}), for each i < h^{32}; (ii) [reduced] the degree of f_{h} in the algebraic variable X_{j} = Z_{i} is strictly

less than the one of f_{i}, deg_{j}(f_{h}) < d_{i} for each i < h;

31 This concept is present in Ritt, op. cit. p.34. 32 As a consequence A is a triangular set; we therefore use the same notation

as above denoting (π_{1}, …, π_{n}) and (L_{0}, …, L_{n}) the corresponding fields and projections.

(iii)[normalized] T_{<}(f_{h}) ∈ k[V_{1}, …, V_{d}][Z_{h}], (compare Remark 42.2.2); (iv) [R_{0}-normalized] lc(f_{h}) ∈ S_{0};

(v) [squarefree] Res(π_{j−1}(f_{h}), π_{j−1}(f′_{h})) ∈ L_{j−1} is invertible^{33}; (vi) [primitive] Cont(π_{j−1}(f_{h})) = 1 in L_{j−1}[X_{j}].

A is called a weak admissible Lazard sequence (or: a regular set) whose associated map is (π_{1}, …, π_{n}) and whose associated tower of simple exten-

sions is (L_{0}, …, L_{n}) if it satisfies only conditions (i-iv).  �

Definition 42.5.3 (Lazard). Let

• A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}] be an admissible Lazard sequence,

• L the ideal generated by A,

• I_{j} the initial of A_{j}, for each j,

• I := ∏_{i=1}^{r} I_{i}.

Then the set

ℨ(A) := Z(A) \ Z({I}) = {α ∈ k^{n} : A_{1}(α) = ⋯ = A_{r}(α) = 0 ≠ I(α)}

is called the quasi-component associated to A and the ideal

Sat(A) := L : I^{∞} := H

is called the quasi-prime ideal associated to A.

Remark 42.5.4 (Lazard). Condition (iii) requires (compare Remark 42.2.2) that, for each h and i, denoting j := class(Lp_{i}(f_{h})) we have X_{j} ∈ {V_{1}, …, V_{d}}. If this is not the case and X_{j} = Z_{ι} for some ι, then, either

• gcd(Lp_{i}(f_{h}), f_{ι}) ≠ 1 and we find a partial factorization of f_{ι} or

• gcd(Lp_{i}(f_{h}), f_{ι}) = 1 = s Lp_{i}(f_{h}) + tf_{ι} for suitable polynomials s, t ∈

k[X_{1}, …, X_{j}] so that F_{h} := sf_{h} is such that class(Lp_{i}(F_{h})) < class(Lp_{i}(f_{h}). �

Remark 42.5.5. If

• the ideal 𝕀(A) ⊂ k[X_{1}, …, X_{n}] is zero-dimensional and we choose R_{0} := L_{0} := k, S_{0} := {1}, or

• we restrict our considerations to its extension 𝕀(A)k(V_{1}, …, V_{d})[Z_{1}, …, Z_{r}] and we choose R_{0} := L_{0} := k(V_{1}, …, V_{d}), S_{0} := {1}

in both cases all variables are algebraic and a set A := {f_{1}, …, f_{r}} where each f_{i} is chosen monic, is an admissible Lazard sequence iff it is an admissible Duval sequence.

In fact, if A is an admissible Lazard sequence, then (v) allows to deduce that, since Res(f_{1}, f′_{1}) ∈ L_{0}, (Proposition 6.6.4) f_{1} is squarefree in L_{0}[X_{1}] and L_{1} is a Duval field (Definition 11.4.2); and, inductively, that

33 As a consequence π_{j−1}(f_{h}) is squarefree.

• L_{j−1} is a direct sum of fields L_{j−1} = ⊕_{ι}L_{j−1 κ}, so that, denoting π_{j−1 κ} : L_{j−1} → L_{j−1 κ} the canonical projection,

• condition (v), Res(π_{j−1}(f_{j}), π_{j−1}(f_{j})′) ∈ L_{j−1}, implies that

Res(π_{j−1 κ}π_{j−1}(f_{j}), π_{j−1 κ}π_{j−1}(f_{j})′) ∈ L_{j−1 κ}

• and that each π_{j−1 κ}π_{j−1}(f_{h}) is squarefree in L_{j−1 κ}[Z_{j}]

• so that L_{j} is a Duval field.

Conversely (i-ii) is satisfied by any admissible sequence and (v) by an admissible Duval sequence, (iii-iv) are equivalent to the requirement that the f_{i}’s are monic, and (vi) is trivially satisfied since we assume S_{0} := {1}. �

Remark 42.5.6 (Lazard). The relation with admissible Ritt sequence is thus expounded by Lazard^{34}:

The notion of [admissible Lazard sequence] is stronger than the notion of characteristic set in the Ritt–Wu Wen-tsün method: Characteristic sets are only subject to conditions (i) and (ii); but we will see that this is not sufficient; in particular a characteristic set may correspond to an empty ”component” of the zero-set. This is avoided by condition (iii). The conditions (iv) to (vi) are needed in order to obtain the unicity of the traiangular set associated with a quasicomponent.

and^{35}

Wu Wen-tsün’s algorithm, like Buchberger’s one, depends on many choices; moreover, the result of Wu Wen-tsün’s algorithm is not uniquely determined. [...] Thus there is a need for a more canonical algorithm, that is an algorithm in which the result (or even better the intermediate results) is more intrinsic, that is, depends on the algebraic structure of the input and not on the algorithm itself. This definition of ”intrinsic” is rather imprecise; it may be better understood by considering the example of an algorithm which is intrinsic, namely the subresultant algorithm, which has the property that the coefficients of the successive remainders may be defined as subdeterminants of Sylvster matrix. [..] For getting a canonical result, [Lazard] strengthen the definition of a triangular set by asking that the polynomials in it are squarefree, primitive and monic in some technical sense. With these conditions, the set of the solutions of an algebraic system is uniquely decomposed

34 Lazard D., A new method for solving algebraic systems of posisitive dimension

Disc. Appl. Math. 33 (1991), p.151. 35 Lazard D. Systems of algebraic equations (algorithms and complexity) Symposia

Mathematica 34 (1993), 84–105, Cambridge Univ. Press .

in so called quasi-components which are themselves in one to one corrispndence with these strengtened triangular systems.

Historical Remark 42.5.7. The existence in Wu’s solver of empty components ℨ(A) := Z(A) \ Z(𝕀(I)) = {α ∈ k^{n} : A_{1}(α) = ⋯ = A_{r}(α) = 0 ≠ I(α)} is illustrated by Example 42.3.7 where for the characteristic set

G := {f_{1}, f_{2}, f_{3}} := {X_{1}^{6} − X_{1}^{4}, (X_{1}^{4} − 2X_{1}^{2})X_{3}, (X_{1}^{2} − 1)X_{2}X_{4} + X_{3}}

we have

I = ∏_{i=1}^{3} I_{i} = (X_{1}^{4} − 2X_{1}^{2})(X_{1}^{2} − 1) = \sqrt{f_{1}} · (X_{1}^{3} − 2X_{1})

so that ℨ(G) = ∅.

This of course cannot happen in the old-fashioned Ritt’s solver where each ’solution’ is the characteristic set of a prime. As I already remarked in Historical Remark 42.4.5, Lazard’s contribution consists in relaxing the notion of ’characteristic sets’ in order to avoid factorization while preserving both the general structure and the relevant properties.

In order to reach this result, it is clearly sufficient to impose condition (iii), which, as we have observed in Remark 42.5.4, can be forced just performing the Extended Euclidean Algorithm to Lp_{i}(f_{h}) and f_{ι} in all cases in which X_{j} = Z_{ι} for j := class(Lp_{i}(f_{h}).

The comparison with the notion of ’solving’ discussed in Section 34.5 is striking: what Lazard did was simply substituting regular sets to admissible sequences, quasi-primes to primes and quasi-components to irreducible vari-

eties!  �

Theorem 42.5.8 (Aubry et al.). Let

A := {f_{1}, …, f_{r}} ⊂ R_{0}[X_{1}, …, X_{n}]

be a weak admissible Lazard sequence whose associated map is (π_{1}, …, π_{n}) and whose associated tower of simple extensions is (L_{0}, …, L_{n}). Then for each p ∈ R_{0}[X_{1}, …, X_{n}], the following conditions are equivalent

(1) π_{n}(p) = 0,

(2) p ∈ Rem(A),

(3) p ∈ Sat(A).

Proof. The proof is by induction by n. We begin by remark that if we set π_{0} the identity on R_{0} and A := ∅ the statements become

(1) p = π_{0}(p) = 0,

(2) p ∈ Rem(∅),

(3) p ∈ Sat(∅) = {0} which are obviously equivalent.

So we can assume n > 0 and that the theorem holds for R_{0}[X_{1}, …, X_{n−1}] and we denote S := Sat(A) ⊂ R_{0}[X_{1}, …, X_{n−1}].

If X_{n} is trascendental, for each p = ∑_{i=0}^{d} a_{i}X_{n}^{i} ∈ R_{0}[X_{1}, …, X_{n−1}][X_{n}] the result follows by induction since we have

(1) π_{n}(p) = 0 ⟺ π_{n−1}(a_{i}) = 0 for each i,

(2) p ∈ Rem(A) ⟺ a_{i} ∈ Rem(A) for each i,

(3) p ∈ Sat(A) ⟺ a_{i} ∈ S for each i.

If, instead X_{n} = Z_{i} is algebraic denote r the remainder of p w.r.t. f_{i} and express it as r = ∑_{l=0}^{d} a_{l}X_{n}^{l} ∈ R_{0}[X_{1}, …, X_{n−1}][X_{n}]. Then the result is a consequence of the following claims:

(1) π_{n}(p) = 0 ⟺ π_{n}(r) = 0;

(2) π_{n}(r) = 0 ⟺ π_{n−1}(r) = 0;

(3) π_{n−1}(r) = 0 ⟺ π_{n−1}(a_{l}) = 0 for each l;

(4) r ∈ Rem(A) ⟺ a_{l} ∈ Rem(A) for each l;

(5) π_{n−1}(r) = 0 ⟺ r ∈ Rem(A);

(6) π_{n}(p) = 0 ⟺ p ∈ Rem(A);

(7) π_{n}(p) = 0 ⟺ p ∈ Sat(A),

whose proof is the following:

(1) By definition π_{n}(Lp(f_{i})) = π_{n−1}(Lp(f_{i})) ∈ L_{n} is a unit, while π_{n}(f_{i}) = 0; therefore from r = Lp(f_{i})^{w}p + qf_{i} we have

π_{n}(r) = π_{n}(Lp(f_{i}))^{w}π_{n}(p) + π_{n}(q)π_{n}(f_{i}) = π_{n}(Lp(f_{i}))^{w}π_{n}(p)

whence the claim.

(2) Since π_{n}(r) = 0 ⟺ π_{n−1}(r) ∈ (π_{n−1}(f_{i})), the claim follows because deg_{n}(r) < deg_{n}(f_{i}) and π_{n−1}(Lp(f_{i})) ∈ L_{n} is a unit.

(3) obvious;

(4) obvious;

(5) by inductive assumption;

(6) by the list of implications and by the remark that the remainders of r and p are the same;

(7) since Rem(A) ⊂ Sat(A), it is sufficient to prove that p ∈ Sat(A) ⟹ π_{n}(p) = 0. We have I^{m}p ∈ 𝕀(A) where m is a suitable integer and I is the product of all initials of the elements in A.

Clearly π_{n}(I) is a unit, so that π_{n}(I^{m}p) = 0 implies π_{n}(p) = 0.  �

Theorem 42.5.9. Let A := {A_{1}, …, A_{r}} ⊂ k[X_{1}, …, X_{n}] be a triangular set generating the ideal L.

Then the following condition are equivalent:

(1) A is a weak admissible Lazard sequence;

(2) A is an admissible Ritt sequence of Sat(A); (3) Sat(A) = Rem(A).

Proof.

(1) ⟹ (3) is Theorem 42.5.8.

(2) ⟺ (3) is a consequence of Lemma 42.1.12.

(3) ⟹ (1) Assume that (3) holds while (1) does not. Inductively we can assume that B := A ∩ R_{0}[X_{1}, …, X_{n−1}] is a weak admissible Lazard sequence whose associated map is (π_{1}, …, π_{n−1}) and whose associated tower of simple extensions is (L_{0}, …, L_{n−1}). Clearly X_{n} = Z_{r} is algebraic and, since (1) does not hold, the initial I_{r} of A_{r} is such that π_{n−1}(I_{r}) is a zero divisor in L_{n−1}, so there is p ∈ k[X_{1}, …, X_{n−1}] such that π_{n−1}(I_{r}p) = 0 and π_{n−1}(p) ≠ 0. Therefore I_{r}p ∈ Sat(B) and the remainder r of p w.r.t. A is not zero. Clearly also I_{r}r ∈ Sat(B) and r ∈ Sat(A). Thus r ∈ k[X_{1}, …, X_{n−1}] satisfies

r ∈ Sat(A) and r ∉ Rem(A) giving the required contradiction.  �

An ideal [Sat(A)] which is the saturated ideal of a [weak admissible Lazard sequence A = {A_{1}, …, A_{r})] is said triangularizable; it is always equi-dimensional of dimension [n − r], where n is the number of variables and [r] the length of [A]. A prime ideal is always triangularizable, and the primes associated to a triangularizable ideal are simply obtained by factoring recursively each [A_{i}] in the field extensions definied by the factors of [A_{1}, …, A_{i−1}]. It should be remarked here that [weak admissible Lazard sequences] are a good alternative to Gröbner base for representing triangularizable and prime ideals in computers: the number of polynomials in a triagular set is always bounded by the number of variables, which is not the case for generating sets of Gröbner bases of prime ideals. Computing a Gröbner basis from a triangular set may be done by any Gröbner base algorithm, and is usually not too difficult. The inverse transformation is very easy for prime ideals. ^{36}

Actually, the standard way for a complete resolution of a polynomial system consists in the following scheme. 1 Compute a lex Gröbner base, either directly or through a change

base ordering. This step checks zero-dimensionality. 2 Deduce from it a set of [weak admissible Lazard sequences]. 3 For each of these triangular systems, compute a RUR [Rational

Univariate Representation]. 4 For each RUR compute a numerical approximation of the solu-

tions together with a bound of the error. ^{37}

36 D. Lazard, Resolution of polynomial systems Proc. ASCM 2000, World Scientific

(2000) 1–8 37 D. Lazard, On the specification for solvers of polynomial systems Proc. ASCM

2001, World Scientific (2001) 1–10

In the next sections we will discuss efficient algorithms to compute triangular sets; the first, due to Lazard returns a weak admissible Lazard sequences and applyies also in the non-zero-dimensional case; the second, by Möller requires zero-dimensionality. In the last section we discuss the notion of RUR and the related algorithms.

42.6 Lazard’s Solver

Let us begin by remarking that, since admissible Lazard sequences and admissible Duval sequences coincide, arithmetical operations in each member L_{i} of a tower of simple extensions, can be performed à la Duval.

In particular:

• when π_{j}(p) ∈ L_{j} — where p ∈ k[X_{1}, …, X_{j}]\k[X_{1}, …, X_{j−1}] and X_{j} = Z_{i} is algebraic — is a zero-divisor, then it is sufficient to compute, in L_{j−1}[Z_{i}], f′ := gcd(π_{j−1}(p), π_{j−1}(f_{i})) and f” := \frac{π_{j−1}(f_{i})}{f′} in order to obain a Duval splitting

L_{j} ≅ L_{j−1}[Z_{i}]/f′ ⊕ L_{j−1}[Z_{i}]/f′

where, denoting π′ : L_{j} → L_{j−1}[Z_{i}]/f′ and π” : L_{j} → L_{j−1}[Z_{i}]/f” the canonical projections, π′π_{j}(p) = 0 and π”π_{j}(p) is invertible;

• testing invertibility of π_{j}(p) ∈ L_{j}, for a polynomial

p ∈ k[X_{1}, …, X_{j}] \ k[X_{1}, …, X_{j−1}]

consists in testing invertibility of

– π_{j−1}(Lp(p)) ∈ L_{j−1} if X_{j} is trascendental,

– Res(π_{j−1}(f_{h}), π_{j−1}(f_{i})) ∈ L_{j−1} if X_{j} = Z_{i} is algebraic;

• computing the inverse of an invertible element π_{j}(p) ∈ L_{j} — where p ∈ k[X_{1}, …, X_{j}] \ k[X_{1}, …, X_{j−1}] and X_{j} = Z_{i} is algebraic — in principle requires to compute gcd(p, f_{i}) in L_{j−1}[X_{j}] but^{38}

two difficulties arise: the first one is that the Euclidean algorithm and its generalizations are defined only for polynomials on integer rings. Fortunately, [Duval’s Model] permits us to compute as if the coefficients were in a field if we split when we encounter a zero-divisor. The second difficulty is to decide which Euclidean algorithm to use: the coefficients being polynomials, an elementary algorithm will generate a swell of coefficients; thus we have to use the subresultant algorithm^{39}; but it needs exact quotients which are not well defined in our context.

38 Lazard D., A new method for solving algebraic systems of posisitive dimension

Disc. Appl. Math. 33 (1991), p. 154 39 id est the version of the Euclidean Algorithm proposed by Collins and Brown

and briefly discussed in Example 1.6.1 and Historical Remarks 1.6.2.

It consists, given two polynomials P_{0}, P_{1} ∈ D[X], where D is a domain, in producing, by means of pseudo-division algorithm, a PRS

P_{0}, P_{1}, …, P_{r} = gcd(P_{0}, P_{1}) ∈ D[X]

We suggest the following approach: apply the subresultant algorithm to the input viewed as multivariate polynomials in [k[X_{1}, …, X_{j}]]; reduce the subresultants, starting from low degrees; the first which does not reduce to zero reduces to a factor of [f_{i}] viewed as a polynomial [in L_{j−1}[X_{j}] = L_{j−1}[Z_{i}]].

• Condition (vi) requires to perform gcd computations over a Duval field L_{j−1}[X_{j}] = L_{j−1}[Z_{i}]; however condition (iv) implies that one of the coefficients of f_{i} is a member of k[V_{1}, …, V_{d}] thus no splitting occurs.

The central procedure of Lazard’s Solver is an algorithm intersect(p, A) where p ∈ k[X_{1}, …, X_{n}] and A ⊂ k[X_{1}, …, X_{n}] is an admissible Lazard sequence, and whose output is a finite family 𝔅 := {B_{1}, …, B_{l}} of admissible Lazard sequences which satisfy

Z(𝕀(p)) ∩ ℨ(A) ⊆ ∪_{i=1}^{l}ℨ(B_{i}) ⊆ \overline{Z(𝕀(p)) ∩ ℨ(A)}.

Given a finite family 𝔄 := {A_{1}, …, A_{l}} of admissible Lazard sequences we denote

intersect(p, 𝔄) := ∪_{i=1}^{l}intersect(p, A_{i})

Finally, given a finite set

G := {g_{1}, …, g_{m}} ⊂ k[X_{1}, …, X_{n}],

the procudere

solve(G) := intersect(g_{1}, intersect(g_{2}, intersect(⋯ intersect(g_{m}, ∅))))

returns, as output, a finite family 𝔅 := {B_{1}, …, B_{l}} of admissible Lazard sequences which satisfy

which satisfes the relations

P_{i+2}/c_{i} = b_{i}P_{i} − Q_{i+1}P_{i+1}

for suitable Q_{i+1} ∈ D[X] and b_{i} ∈ D, and predictable elements c_{i} ∈ D. These data allow also to compute (essentially as in Proposition 1.3.1) polynomials S_{i}, T_{i} satisfying the Bezout’s Identities P_{i} = P_{0}S_{i} + P_{1}T_{i}.

In the quoted passage, the proposed approach is to apply the algorithm to π_{j−1}(f_{i}) and π_{j−1}(p) in L_{j−1}[X_{j}] = L_{j−1}[Z_{i}], where p ∈ k[X_{1}, …, X_{j}] \ k[X_{1}, …, X_{j−1}] and X_{j} = Z_{i} is algebraic. The technical problem consists that the computation requires zero-testing; the proposal solution requires to

– compute a PRS f_{i}, p, P_{2}, …, P_{r} of f_{i} and p in k[X_{1}, …, X_{j−1}][X_{j}],

– evaluate π_{j−1}(P_{r}), π_{j−1}(P_{r−1}), … until a non-zero element π_{j−1}(P_{ρ}) is produced which therefore satisfies

π_{j−1}(P_{ρ}) = gcd(π_{j−1}(f_{i}), π_{j−1}(p)) ∈ L_{j−1}[X_{j}] = L_{j−1}[Z_{i}].

Z(G) = ∩_{i}Z(𝕀(g_{i}))
= Z(𝕀(g_{1})) ∩ (Z(𝕀(g_{2})) ∩ (⋯ (Z(𝕀(g_{m})) ∩ ℨ(∅))))
= ∪_{i=1}^{l}ℨ(B_{i}).

Algorithm 42.6.1 (Lazard). intersect(p, A) applies another procedure

(r) := normalize(p, A)

whose input is the polynomial p ∈ k[X_{1}, …, X_{n}] and the admissible Lazard sequence

A := {f_{1}, …, f_{r}} ⊂ R_{0}[X_{1}, …, X_{n}] = R_{0}[V_{1}, …, V_{d}][Z_{1}, …, Z_{r}]

w.r.t. which we use the same notation as in Section 42.5^{40} and which computes two polynomials q and r such that

(1) π_{n}(qp) = π_{n}(r),

(2) π_{n}(p) = 0 ⟺ π_{n}(r) = 0 and

(3) r is reduced, normalized and R_{0}-normalized.

Here is the procedure:

set q := 1,

[reduced] we compute the remainder r of p w.r.t. A, [normalized] while

T_{<}(r) := X_{j_{ι}}^{δ_{ι}}X_{j_{ι−1}}^{δ_{ι−1}} ⋯ X_{j_{1}}^{δ_{1}}X_{j_{0}}^{δ_{0}} ∉ k[V_{1}, …, V_{d}], δ_{ι} ≠ 0

then

• compute (compare Remark 42.5.4) a polynomial^{41} s ∈ k[X_{1}, …, X_{j_{ι}}] for which class(Lp_{ι}(sr) < class(Lp_{ι}(r),

• compute the remainder r of sr w.r.t. A and

• set q := sq, [R_{0}-normalized] if lc(r) ∉ S_{0} choose^{42} c ∈ R_{0} such that lc(cr) ∈ S_{0} and set

r := cr, q := cq.  �

Algorithm 42.6.2 (Lazard). We can now present the procedure

𝔅 := intersect(p, A)

where A is the admissible Lazard sequence

40 In particular {V_{1}, …, V_{d}} (respectively {Z_{1}, …, Z_{r}}) is the set of the trascen-

dental (respectively: algebraic) variables for A. 41 Here a Duval splitting could happen. 42 If we have R_{0} := L_{0} := k and S_{0} := {1}, this simply requires to choose c :=

lc(r)^{−1}.

A := {f_{1}, …, f_{r}} ⊂ R_{0}[X_{1}, …, X_{n}] = R_{0}[V_{1}, …, V_{d}][Z_{1}, …, Z_{r}]

and we use the same notation as in Section 42.5^{43}:

(1) 𝔅 := ∅, (r) := normalize(p, A)

(2) If • r = 0: set 𝔅 := {A} and exit;

• r ∈ k \ {0}: set 𝔅 := ∅ and exit;

• r ∉ k: goto (3);

(3) expressing r as

r = Lp(r)X_{j}^{δ} + r

where j = class(r), Lp(r) ∈ k[X_{1}, …, X_{j−1}], deg_{j}(r) < δ = deg_{j}(r), set

𝔅 := 𝔅 ∪ intersect(r, intersect(Lp(r), A)).

(4) Compute Cont(r) ∈ k[X_{1}, …, X_{j−1}][X_{j}] and set r := \frac{r}{Cont(r)}, thus forcing r to be primitive.

(5) Setting j = class(r), denote i, δ the values such that • A ∩ R_{0}[X_{1}, …, X_{j}] = {f_{1}, …, f_{i}}

• class(f_{i}) < j

• {V_{1}, …, V_{δ}} = {X_{1}, …, X_{j}} ∩ {V_{1}, …, V_{d}}

• X_{j} = V_{δ} is trascendental, and set

A_{j}^{−} := {f_{1}, …, f_{i}}, A_{j}^{+} := {f_{i+1}, …, f_{r}}.

(6) Compute R := Res(π_{j−1}(r), π_{j−1}(r′)) ∈ L_{j−1}.

• If R is invertible, so that π_{j−1}(r) is squarefree in L_{j−1}[X_{j}], goto (7)

• If, instead, R is not invertible, then

– compute^{44}, using the subresultant algorithm, a factor r_{0} | r such that, in L_{j−1}[X_{j}],

π_{j−1}(r_{0}) = gcd(π_{j−1}(r), π_{j−1}(r′)),

– set r := r/r_{0}

– goto (3)

(7) If A_{j}^{+} = ∅ set ℭ := A_{j}^{−} ∪ {r}

(8) If A_{j}^{+} ≠ ∅ • compute ℭ := intersect(f_{i+1}, intersect(⋯ intersect(f_{r}, A_{j}^{−}∪{r}))).

• Set ℭ := {C ∈ ℭ : normalize(f_{l}, C^{−}) ≠ 0 for each l, i < l ≤ r}^{45}.

(9) 𝔅 := 𝔅 ∪ intersect(p, ℭ).  �

43 In particular {V_{1}, …, V_{d}} (respectively {Z_{1}, …, Z_{r}}) is the set of the trascen-

dental (respectivelty: algebraic) variables for A. 44 This can produce a Duval splitting. 45 Where C^{−} = C ∪ R_{0}[X_{1}, …, X_{j}], j = class(r).

170 42. Lazard II

Example 42.6.3. Let us compute solve(G) where (compare Example 42.3.7)

G := {f_{1}, f_{2}, f_{3}} ⊂ k[X_{1}, X_{2}, X_{3}, X_{4}]

and

f_{1} := X_{1}^{6} − X_{1}^{4}, f_{2} := (X_{1}^{4} − 2X_{1}^{2})X_{3}, f_{3} := (X_{1}^{2} − 1)X_{2}X_{4} + X_{3} We have

normalize(f_{3}, ∅) = f_{3},

C_{1} := {f_{3}} = {(X_{1}^{2} − 1)X_{2}X_{4} + X_{3}}, normalize(f_{3}, C_{1}^{−}) = f_{3}, intersect(f_{3}, ∅) = {C_{1}} =: ℭ_{1}, normalize(f_{2}, C_{1}) = f_{2}

r_{1} := Lp(f_{2}) = X_{1}^{4} − 2X_{1}^{2} normalize(r_{1}, C_{1}) = r_{1};

r_{2} := \sqrt{r_{1}} = X_{1}^{3} − 2X_{1} C_{2} := C_{1}^{−} ∪ {r_{2}} = {r_{2}} = {X_{1}^{3} − 2X_{1}} normalize(f_{3}, C_{2}) = X_{2}X_{4} + X_{3}(X_{1}^{2} − 1) =: r_{3}^{46},

r_{4} := Lp(r_{3}) = X_{2}, r_{5} := r_{3} − r_{4}X_{4} = (X_{1}^{2} − 1)X_{3}

normalize(r_{4}, C_{2}^{−}) = r_{4}
C_{3} := C_{2}^{−} ∪ {r_{4}} = {r_{2}, r_{4}} = {X_{1}^{3} − 2X_{1}, X_{2}},
intersect(r_{4}, {C_{2}}) = {C_{3}}
normalize(r_{5}, C_{3}^{−}) = X_{3} =: r_{6}^{47},

C_{4} := C_{3}^{−} ∪ {r_{6}} = {r_{2}, r_{4}, r_{6}} = {X_{1}^{3} − 2X_{1}, X_{2}, X_{3}},

normalize(r_{5}, C_{3}^{−}) = r_{5},
intersect(r_{5}, {C_{3}}) = {C_{4}}

C_{5} := C_{2} ∪{r_{3}} = {r_{2}, r_{3}} = {X_{1}^{3} −2X_{1}, X_{2}X_{4} +X_{3}(X_{1}^{2} −1)}

ℭ_{2} := {C_{4}, C_{5}}

intersect(f_{3}, C_{2}) = ℭ_{2} normalize(r_{1}, C_{i}^{−}) ≠ 0, i ∈ {4, 5}

intersect(r_{1}, ℭ_{1}) = ℭ_{2} \frac{f_{2}}{Cont(f_{2})} = X_{3} = r_{6}, C_{6} := C_{1}^{−} ∪ {r_{6}} = {r_{6}} = {X_{3}} normalize(f_{3}, C_{6}) = (X_{1}^{2} − 1)X_{2}X_{4} =: r_{7}

Lp(r_{7}) = (X_{1}^{2} − 1)X_{2} =: r_{8} normalize(r_{8}, C_{6}) = r_{8}

Lp(r_{8}) = (X_{1}^{2} − 1) =: r_{9} normalize(r_{9}, C_{6}) = r_{9}

C_{7} := C_{6}^{−} ∪ {r_{9}} ∪ C_{6}^{+} = {r_{9}, r_{6}} = {X_{1}^{2} − 1, X_{3}} intersect(r_{9}, {C_{6}}) = {C_{7}} \frac{r_{8}}{Cont(r_{8})} = X_{2} = r_{4}, C_{8} := C_{6} ∪ {r_{4}} = {r_{4}, r_{6}} = {X_{2}, X_{3}}

intersect(r_{8}, {C_{6}}) = {C_{7}, C_{8}}

46 We have (X_{1}^{2} − 1)f_{3} − X_{1}X_{2}X_{4}r_{2} = X_{2}X_{4} + X_{3}(X_{1}^{2} − 1). 47 We have (X_{1}^{2} − 1)r_{5} − X_{1}X_{3}r_{2} = X_{3}.

\frac{r_{7}}{Cont(r_{7})} = X_{4} = r_{10},
C_{9} := C_{6}^{−} ∪ {r_{10}} = {r_{6}, r_{10}} = {X_{3}, X_{4}}

intersect(f_{3}, {C_{6}}) = {C_{9}} ℭ_{2} := {C_{4}, C_{5}, C_{7}, C_{8}, C_{9}} normalize(f_{2}, C_{i}^{−}) ≠ 0, i ∈ {4, 5, 7, 8, 9} intersect(f_{2}, ℭ_{1}) = intersect(f_{2}, C_{1}) = ℭ_{2} Rem(f_{1}, r_{2}) = 2X_{1}^{2} := r_{11}, by Duval splitting we get

r_{12} := X_{1},

normalize(r_{11}, {r_{12}}) = r_{12},
normalize(r_{4}, {r_{12}}) = r_{4},
normalize(r_{6}, {r_{12}, r_{4}}) = r_{6}
C_{11} = {r_{12}, r_{4}, r_{6}} = {X_{1}, X_{2}, X_{3}},
r_{13} := X_{1} − 2
normalize(r_{11}, {r_{13}}) = 4

intersect(f_{1}, C_{4}) = C_{11}, Rem(f_{1}, r_{2}) = 2X_{1}^{2} := r_{11}, by Duval splitting we get

r_{12} := X_{1},

normalize(r_{11}, {r_{12}}) = r_{12}
normalize(r_{3}, {r_{12}}) = X_{2}X_{4} − X_{3} =: r_{14},
C_{12} := {r_{11}, r_{14}} = {X_{1}, X_{2}X_{4} − X_{3}}
r_{13} := X_{1} − 2
normalize(r_{11}, {r_{13}}) = 4

intersect(f_{1}, C_{5}) = C_{12} normalize(f_{1}, C_{7}) = 0, normalize(f_{1}, C_{8}) = f_{1}, \sqrt{f_{1}} = X_{1}^{3} − X_{1} =: r_{15}, intersect(f_{1}, C_{8}) = {r_{15}, r_{4}, r_{6}} = {X_{1}^{3} − X_{1}, X_{2}, X_{3}} =: C_{13} normalize(f_{1}, C_{9}) = f_{1}, \sqrt{f_{1}} = X_{1}^{3} − X_{1} =: r_{15}, intersect(f_{1}, C_{9}) = {r_{15}, r_{6}, r_{10}} = {X_{1}^{3} − X_{1}, X_{3}, X_{4}} =: C_{14} solve(G) = intersect(f_{1}, ℭ_{2}) = ℭ_{3} := {C_{11}, C_{12}, C_{7}, C_{13}, C_{14}}

so that

\begin{aligned}Z(C_{11}) & = & Z(\{X_{1}, X_{2}, X_{3}\}) & = & \{(0, 0, 0, a), a ∈ k, \}, \\ Z(C_{12}) & = & Z(\{X_{1}, X_{2}X_{4} − X_{3}\}) & = & \{(0, a, b, \frac{b}{a}), a, b ∈ k, a ≠ 0\}, \\ Z(C_{7}) & = & Z(\{X_{1}^{2} − 1, X_{3}\}) & = & \{(x, a, 0, b), x ∈ \{1, −1\}, a, b ∈ k, \}, \\ Z(C_{13}) & = & Z(\{X_{1}^{3} − X_{1}, X_{2}, X_{3}\}) & = & \{(x, 0, 0, a), x ∈ \{0, 1, −1\}, a ∈ k, \}, \\ Z(C_{14}) & = & Z(\{X_{1}^{3} − X_{1}, X_{3}, X_{4}\}) & = & \{(x, a, 0, 0), x ∈ \{0, 1, −1\}, a ∈ k, \}, \end{aligned}

where

Z(C_{13}) ∪ Z(C_{14}) ⊂ Z(C_{12}) ∪ Z(C_{7}).

Algorithm 42.6.4 (Lazard). For to removing redundant components, Lazard proposes an algorithm inclusion?(T, U) which is performed to the set of the quasi-components ordered by increasing dimension and in which each quasicomponent T is compared with each component U of higher dimension to test whethere T ⊂ U.

The procedure consists in checking whether normalize(f, T) = 0 for each f ∈ U, the answer being positive iff all tests have success.

Of course the tests produce a Duval-splitting in T.  �

Example 42.6.5. For instance, with the present example, the tests

inclusion?(Z(C_{i}), Z(C_{12})), i ∈ {13, 14},

return the splittings

Z(C_{13}) = Z(C_{7}) ∪ Z(C′_{13}),  C′_{13} = Z({X_{1}, X_{2}, X_{3}}) = {(0, 0, 0, a), a ∈ k, }
Z(C_{14}) = Z(C_{7}) ∪ Z(C′_{14}),  C′_{14} = Z({X_{1}, X_{2}, X_{3}}) = {(0, a, 0, 0), a ∈ k, }

the answer being positive for the components Z(C′_{i}).

42.7 Ritt bases and Gröbner bases

Let k be a field of characteristic zero, P := k[X_{1}, …, X_{n}],

T := {X_{1}^{a_{1}} ⋯ X_{n}^{a_{n}} : (a_{1}, …, a_{n}) ∈ ℕ^{n}},

< be the lexicographical ordering on T induced by X_{1} < … < X_{n}.

Let I ⊂ P and let G := {g_{1}, …, g_{s}} be the reduced Gröbner basis of I ordered so that

T(g_{1}) < T(g_{2}) < … < T(g_{s−1}) < T(g_{s});

and denote, for each i, 1 ≤ i ≤ n, G_{i} := G ∩ k[X_{1}, …, X_{i}].

Adapting Definition 42.5.1, we say that a variable X_{i} is called

algebraic for I if there is g ∈ G_{i} \ G_{i−1} trascendental for I if G_{i} = G_{i−1}.

As usual we relabel the variables as

k[X_{1}, …, X_{n}] ≅ k[V_{1}, …, V_{d}, Z_{1}, …, Z_{r}]

so that {V_{1}, …, V_{d}} (respectively {Z_{1}, …, Z_{r}}) is the set of the trascendental (respectively: algebraic) variables for G.

To each reduced lex Gröbner basis G we associate a set M(G) inductively (on the rank r := r(𝕀(G)) of the ideal generated by G as follows:

if r = 1, then set M(G) := {g_{1}}, if r > 1, denoting i the value for which Z_{r} = X_{i}, so that G_{i} \ G_{i−1} ≠ ∅ and

(as we will prove below) M(G_{i−1}) is a triangular set, then we set

• M(G) := M(G_{i−1}), if G_{i} ⊂ Rem(M(G_{i−1}));

• if, instead, G_{i} ⊄ Rem(M(G_{i−1}) then we set M(G) := M(G_{i−1}) ∪ {g_{j}} where j is the minimal value for which the remainder of g_{i} w.r.t. M(G_{i−1}) is not zero.

Definition 42.7.1 (Aubry et al.). The set M(G) defined above is called the median set of I, where I is the ideal generated by the lex reduced Gröbner

basis G.  �

Example 42.7.2. For

G := {X_{1}X_{2}, X_{2}X_{3}, X_{3}X_{4}} ∈ k[X_{1}, X_{2}, X_{3}, X_{4}] ≅ k[V_{1}][Z_{1}, Z_{2}, Z_{3}]

we set

M(G_{2}) = {X_{1}X_{2}}, M(G_{3}) = M(G_{2}) since X_{2}X_{3} ∈ Rem(M(G_{2})),

M(G) = M(G_{2}) ∪ {X_{3}X_{4}} = {X_{1}X_{2}, X_{3}X_{4}}.  �

Proposition 42.7.3 (Aubry et al.). Let G ⊂ P be a reduced lex Gröbner basis generating an ideal I and let M(G) be its medial set. Then

(1) M(G) is a non empty triangular set;

(2) M(G) ⊆ I ⊆ Rem(M(G));

(3) M(G) is a fine triangular set;

(4) M(G) is an admissible Ritt sequence;

(5) M(G) is initially reduced.

Proof.

(1) Obvious,

(2) The only non trivial result is the inclusion I ⊆ Rem(M(G)). Assume that there is f ∈ I for which f ∉ Rem(M(G)) and denote r its remainder w.r.t. M(G) remarking that r ∈ I. Therefore there is g ∈ G such that T(g) | T(r). Since r (and so also T(r)) is reduced, the same is true for T(g). Let i := class(g), and A := {h ∈ M(G) : class(h) < i}. Remark that either

• G_{i} ⊂ Rem(A) and A = M(G_{i}), or

• G_{i} ⊄ Rem(A) and there is h ∈ G_{i} such that {h} = M(G_{i}) \ A so that there are three cases: either

(a) g ∈ Rem(A), or^{48}

(b) g = h ∈ M(G) \ A or

(c) g ≠ h ⟹ T(g) > T(h) ^{49}

48 g ∉ Rem(A) so that G_{i} ⊄ Rem(A) and M(G_{i}) \ A = {h}. 49 We assume lc(f) = 1 for each f ∈ G so that T(g) = T(h) ⟹ g = h.

but all these cases reduce to a contradiction:

(a) contradicts the assumption that T(g) is reduced;

(b) ditto;

(c) cannot hold for the same reason: in fact, class(h) = i = class(g) and T(g) > T(h) implie deg_{i}(T(g)) ≥ deg_{i}(T(h)) = deg_{i}(h) which in turn implies that T(g) is reduced by h. �

(3) Assume that for some g ∈ M(G) the remainder of Lp(g) w.r.t.

A := {h ∈ M(G) : T(h) < T(g)} = {h ∈ M(G) : class(h) < class(g)}

is zero, so that, denoting j := class(g) and g′ ∈ k[X_{1}, …, X_{j}] the polynomial such that g = Lp(g)X_{j}^{deg_{j}(g)} + g′, deg_{j}(g′) < deg_{j}(g), the remaider r of g and the one of g′ w.r.t. A are the same. In particular r ∈ I ⊆ Rem(M(G)) whence r = 0, and g ∈ Rem(A) contradicting the construction of M(G).

(4) If follows form (2) and Proposition 42.4.7.

(5) If not, there is a smallest (w.r.t. j := class(g)) element g ∈ G which is not initially reduced w.r.t.

A := {h ∈ M(G) : T(h) < T(g)}
= {h ∈ M(G) : class(h) < j}
=: {A_{1}, …, A_{ρ}}.

On the other side the remainder of Lp(g) is not zero w.r.t. A. Then, Theorem 42.1.18 implies that, for suitable integers w_{i}, and denoting I_{i} the initilal of A_{i}

I_{1}^{w_{1}} ⋯ I_{r}^{w_{r}}g = I_{1}^{w_{1}} ⋯ I_{r}^{w_{r}} Lp(g)X_{j}^{deg_{j}(g)} + I_{1}^{w_{1}} ⋯ I_{r}^{w_{r}}g′

reduces w.r.t. A to a polynomial t := RX_{j}^{deg_{j}(g)} + g′′ ∈ I which is

• reduced w.r.t. A,

• class(t) = j, deg_{j}(t) = deg_{j}(g),

• T(t) = T(R)X_{j}^{deg_{j}(g)} < T(Lp(g))X_{j}^{deg_{j}(g)} = T(g). Then, necessarily, T(t) is divided by T(h) for some h ∈ G_{j} for which Lp(h) ∈ Rem(A); therefore T(h) ∈ Rem(A) and T(t) ∈ Rem(A) contra-

dicting the assumption that t is reduced.  �

It is possible to recover Ritt’s Corollary 42.3.5 as follows: for each algebraic variable Z_{i} = X_{j} denote A_{i} the smallest^{50} polynomial in G_{j}\G_{j−1} and denote A(G) := {A_{1}, …, A_{r}}.

With this notation we have:

50 Recall that the elements in G are enumerated so that T(g_{i}) < T(g_{i+1}) so the ’smallest’ polynomial in G_{j} \ G_{j−1} is also the polynomial g ∈ G_{j} \ G_{j−1} having the <-minimal value T(g).

Theorem 42.7.4. If G generates a prime ideal I, then:

(1) M(G) = A(G),

(2) I = Rem(M(G)) = Sat(M(G)),

(3) \overline{ℨ(M(G))} = Z(I);

(4) ℨ(M(G)) ≠ ∅.

Proof.

(1) It is sufficient to show that for each j the remainder of the initial Lp(A_{j}) w.r.t. B := {A_{1}, …, A_{j−1}} is not zero. If it were zero, we would get a contradiction from Lp(A_{j}) ∈ I which is impossible since A_{j} is a member of a reduced Gröbner basis.

(2) Since I ⊂ Rem(M(G)) by the Proposition above, the result follows from Corollary 42.3.5.

(3) Again by Corollary 42.3.5.(4).

(4) Z(I) ≠ ∅.  �

Example 42.7.5. If G generates just a radical ideal I, it could happen that ℨ(M(G)) = ∅.

Let

F := {X_{1}^{2} − 2, X_{2}^{2} − 2, (X_{1} − X_{2})X_{3}, (X_{1} + X_{2})X_{4}} ∈ k[X_{1}, X_{2}, X_{3}, X_{4}].

The Gröbner basis is G := F ∪ {X_{3}X_{4}} and M(G) = F. Clearly the product of the initials

(X_{1} − X_{2})(X_{1} + X_{2}) = X_{1}^{2} − X_{2}^{2} ∈ (X_{1}^{2} − 2, X_{2}^{2} − 2)

Thus ℨ(M(G)) = ∅.

42.8 Möller’s Zero-dimensional Solver

Lemma 42.8.1. Let J ⊂ Q be a zero-dimensional ideal and let h ∈ Q. It holds

Z(J : h^{∞}) = {α ∈ Z(J) : h(α) ≠ 0}.

Proof. Let us consider the irredeundant primary decomposition J = ⋃_{i=1}^{r} 𝔮_{i} where, for each i, 𝔭_{i} denotes the associate maximal 𝔭_{i} = \sqrt{𝔮}_{i}.

We have (Theorem 26.3.2 (19)) J : h^{∞} = ⋃_{i=1}^{r} 𝔮_{i} : h^{∞} and (Corol-

lary 27.2.12) 𝔮_{i} : h^{∞} = \{ \begin{matrix}Q & iff h ∈ 𝔭_{i} \\ 𝔮_{i} & iff h ∉ 𝔭_{i} \end{matrix} whence the claim follows easily. � Proposition 42.8.2 (Möller). Let J ⊂ Q be a zero-dimensional ideal and let H := {h_{1}, …, h_{t}} ⊂ Q be a set of polynomials such that

Z(H) ⊂ Z(J).

Denoting J_{t} := J and J_{i} := J + 𝕀(h_{i+1}, …, h_{t}), 1 ≤ i < t, it holds

Z(J) = Z(H) ⨆ ⨆_{i=1}^{t} Z(J_{i} : h_{i}^{∞}).

Proof. Clearly

Z(J) \ Z(H) = {α ∈ Z(J) : exists i, i ≤ t, h_{i}(α) ≠ 0}
= ⨆_{i=1}^{t}{α ∈ Z(J) : h_{t}(α) = ⋯ = h_{i+1}(α) = 0 ≠ h_{i}(α)}

= ⨆_{i=1}^{t} Z(J_{i} : h_{i}^{∞})

the last equality following from Lemma 42.8.1.  �

The intendend application of Proposition 42.8.2 requires efficient algorithms in order to compute, given a zero-dimensional ideal J ⊂ Q and a polynomial h ∈ Q \ {0} both J + (h) and J : h^{∞}; different techniques are discussed in Sections 26.3–7. In connection, Möller anticipated some version of Caboara–Traverso ideas (Section 26.6); in particular he stated

Lemma 42.8.3 (Möller). Let 𝔞 = 𝕀(g_{1}, …, g_{s}) ⊂ Q and h ∈ Q\{0}. Then,

M := {(u, v) ∈ Q^{2} : u − hv ∈ 𝔞}

is a module with basis F := {(g_{i}, 0), 1 ≤ i ≤ s} ∪ {(h, 1)}.

Moreover, fixing any termordering ≺ on W and denoting

• {e_{1}, e_{2}} the canonical basis of Q^{2},

• W^{(2)} = {τe_{i},: τ ∈ W, i ∈ {1, 2}},

• ≺_{2} the ≺-compatible termordering on W^{(2)} defined by

τe_{i} ≺_{2} τ′e_{j} ⟺ \{ \begin{matrix}i > j & or \\ i = j & and τ ≺ τ′, \end{matrix}

• G the Gröbner basis of M wrt ≺_{2},

• G_{0} := {b ∈ Q : (0, b) ∈ G},

• G_{1} := {a ∈ Q : (a, b) ∈ G}.

Then (𝔞 : h) = 𝕀(G_{0}) and 𝔞 + (h) = 𝕀(G_{1}).

Proof. Obviously F ⊂ M. For each (u, v) ∈ M there are f_{i} ∈ Q such that u − hv = ∑_{i} f_{i}g_{i} so that (u, v) = ∑_{i} f_{i}(g_{i}, 0) + v(h, 1); this proves M = 𝕀(F).

The relation 𝕀(G_{1}) = 𝕀(g_{1}, …, g_{s}, h) = 𝔞 + (h) is obvious.

The other claim is a direct consequence of the trivial equivalence

(0, v) ∈ M ⟺ −hv ∈ 𝔞 ⟺ v ∈ (𝔞 : h).

�

In order to deduce 𝔞 : h^{∞}, Möller proposed to iteratively apply the same algorithm in order to iteratively deduce 𝔞 : h^{i}; the result is then obtained at stabilization^{51}.

Remark 42.8.4. If, as it is assumed, J is zero-dimensional, Möller also proposes to apply Traverso’s Algorithm 29.3.8 in order to deduce J + (h) and a proper variation of the FGLM algorithm in order to compute (J : h).

Namely, denoting {τ_{1}, …, τ_{u}} = N(J), and, for each f ∈ Q

Rep(f, N(J)) := (γ(f, τ_{1}, N(J)), …, γ(f, τ_{u}, N(J)))

its Gröbner description, in order to obtain (J : h), Möller’s Algorithm is to be applied to the functionals

ℓ_{i} : Q → k : f ↦ γ(fh, τ_{i}, N(J))

in the same way in which FGLM Algorithm is obtained by applying Möller’s to the functionals

ℓ_{i} : Q → k : f ↦ γ(f, τ_{i}, N(J)).

�

Let J ⊂ Q be a zero-dimensional ideal and let G := {g_{1}, …, g_{s}}, lc(g_{i}) = 1, be its Gröbner basis with respect the lex ordering induced by Z_{1} < Z_{2} < … < Z_{r} ordedered so that T(g_{1}) < T(g_{2}) < … < T(g_{s}).

The assumption that the ideal is zero-dimensional trivially implies that g_{s} ∈ K[Z_{1}, …, Z_{r}]\K[Z_{1}, …, Z_{r−1}] and that deg_{r}(g_{i}) := d_{i} < d_{s} := deg_{r}(g_{s}) for each i < s.

As a consequence (Compare Kalkbener’s Theorem 26.5.4)^{52} {Lp(g_{i}), 1 ≤ i < s} is a Gröbner basis w.r.t. <; moreover, since J is zero-dimensional we also have Lp(g_{s}) ∈ k.

Theorem 42.8.5 (Möller). With the present notation we have

𝕀(g_{1}, …, g_{s−1}) : g_{s} = 𝕀(Lp(g_{1}), … Lp(g_{s−1})).

51 Compare the discussion after Lemma 26.3.9. 52 Apparently Kalkbener’s Theorem 26.5.4 and the weaker Möller’s Theorem 42.8.5

are independent.

Proof. If h ∈ J and T(h) < T(g_{s}), then there is j < s such that T(g_{j}) | T(h) and there are c ∈ k \ {0}, τ ∈ W, such that h′ := h − cτg_{j} satisfies h′ ∈ J and T(h′) < T(h) < T(g_{s}).

Thus for each h ∈ J, for which T(h) < T(g_{s}), it holds h ∈ 𝕀(g_{1}, …, g_{s−1}). For each i, 1 ≤ i < s, set h_{i} := Lp(g_{i})g_{s} − Z_{r}^{d_{s}−d_{i}}g_{i}; since

T(Lp(g_{i}))T(g_{s}) = T(Lp(g_{i})Z_{r}^{d_{s}} = T(Lp(g_{i})Z_{r}^{d_{i}})Z_{r}^{d_{s}−d_{i}} = T(g_{i})Z_{r}^{d_{s}−d_{i}}

we have T(h_{i}) < T(g_{s}) and, since h_{i} ∈ J, we have h_{i} ∈ 𝕀(g_{1}, …, g_{s−1}) whence Lp(g_{i})g_{s} ∈ 𝕀(g_{1}, …, g_{s−1}) and Lp(g_{i}) ∈ 𝕀(g_{1}, …, g_{s−1}) : g_{s}. We have thus proven the inclusion 𝕀(Lp(g_{1}), … Lp(g_{s−1})) ⊆ 𝕀(g_{1}, …, g_{s−1}) : g_{s}.

Conversely, let us consider a polynomial g ∈ 𝕀(g_{1}, …, g_{s−1}) : g_{s}, g ≠ 0. Since gg_{s} ∈ 𝕀(g_{1}, …, g_{s−1}), there is i < s such that T(g_{i}) | T(gg_{s}) =

T(g)Z_{r}^{d_{s}} and there are c ∈ k \ {0}, τ ∈ W ∩ k[Z_{1}, …, Z_{r−1}] such that

T(gg_{s}) = cτZ_{r}^{d_{s}−d_{i}}T(g_{i}) = cτT(Lp(g_{i}))Z_{r}^{d_{s}} = cτT(Lp(g_{i})g_{s}).

Denoting g′ := g − cτ Lp(g_{i}) and remarking that^{53}

g′ ∈ (𝕀(g_{1}, …, g_{s−1}) : g_{s}) + 𝕀(Lp(g_{1}), … Lp(g_{s−1})) ⊆ 𝕀(g_{1}, …, g_{s−1}) : g_{s}

we have that either

• T(g′) < T(g) and g − g′ ∈ 𝕀(Lp(g_{1}), … Lp(g_{s−1})) or

• g′ = 0 and g ∈ 𝕀(Lp(g_{1}), … Lp(g_{s−1})).

Thus, by <-indiction we can deduce that

𝕀(g_{1}, …, g_{s−1}) : g_{s} ⊆ 𝕀(Lp(g_{1}), … Lp(g_{s−1}).

�

Corollary 42.8.6 (Möller). With the present notation and setting H := {Lp(g_{i}), 1 ≤ i < s} ∪ {g_{s}} we have

Z(H) ⊂ Z(J).

�

Algorithm 42.8.7 (Möller). With the present notation, the algorithm described in Figure 42.1 produces a triangular set decomposition of the zerodimensional ideal J.

In fact 𝔗 is obtained, according Proposition 42.8.2 by the disjoint union of

53 Recall that we have just proved the inclusion

𝕀(Lp(g_{1}), … Lp(g_{s−1}) ⊆ 𝕀(g_{1}, …, g_{s−1}) : g_{s}.

Fig. 42.1. Möller’s Algorithm

𝔗 := Solve(J) where

J ⊂ Q is a zero-dimensional ideal, < is the lex ordering induced by Z_{1} < … < Z_{r} {g_{1}, …, g_{s}}, lc(g_{i}) = 1, T(g_{1}) < … < T(g_{s}) is the reduced Gröbner basis of J wrt <. 𝔗 = {𝔱_{1}, …, 𝔱_{v}} is a finite set of triangular sets such that

Z(J) = ⨆_{j=1}^{v} Z(𝕀(𝔱_{j}).

Let G be the reduced Gröbner basis wrt < of 𝕀(Lp(g_{1}), … Lp(g_{s−1})); 𝔘 := Solve(𝕀(Lp(g_{1}), …, Lp(g_{s−1}))) 𝔗′ := {𝔱 ∪ {NF(g_{s}, 𝔱)} : 𝔱 ∈ 𝔘} i = 1, G_{s−1} := {g_{1}, …, g_{s}}; While Lp(g_{s−i}) ∉ J do

Compute a reduced Gröbner basis G′_{s−i} of 𝕀(G_{s−i}) : Lp(g_{s−i})^{∞}, Compute a reduced Gröbner basis G_{s−i−1} of 𝕀(G_{s−i}) + 𝕀 (Lp(g_{s−i})), 𝔗_{i} := Solve(𝕀(G′_{s−i})) 𝔗 := 𝔗′ ∪ 𝔗_{i} i := i + 1

• 𝔗′ which gives the triangular set decomposition of 𝕀(H), and

• 𝔗_{i} which gives the triangular set decomposition of J_{i} : Lp(g_{i})^{∞}, for i, l < i < s^{54} where l is the value for which {g_{1}, …, g_{l}} = H ∩ K[Z_{1}, …, Z_{r−1}].

The correctness of the While-loop is a direct consequence of the ordering of the basis elements^{55} and on the fact that

Lp(g_{s−i}) ∈ J ⟹ (𝕀(G_{s−i} : Lp(g_{s−i}^{∞}) = Q ⟺ Z (𝕀(G_{s−i} : Lp(g_{s−i}^{∞})) = ∅.

�

Example 42.8.8. To illustrate the algorithm let us consider Example 39.2.3 where, denoting

54 Remark that Proposition 42.8.2(6) apparently requires, before the While-loop,

to compute

– a reduced Gröbner basis G′_{s} of 𝕀(g_{1}, …, g_{s}) : g_{s}^{∞}),

– a reduced Gröbner basis G_{s−1} of 𝕀(g_{1}, …, g_{s}) + 𝕀(g_{s}),

– the triangular set decomposition of 𝕀(G′_{s})

but this computation is trivilal and returns G′_{s} = {1} and G_{s−1} = {g_{1}, …, g_{s}}. 55 For each l ≥ i we have

Lp(g_{s−i}) ∈ J ⟺ g_{s−i} ∈ k[Z_{1}, …, Z_{r−1}]
⟹ g_{s−l} ∈ k[Z_{1}, …, Z_{r−1}]
⟺ Lp(g_{s−l}) ∈ J

H_{1} := {Z_{1}^{2} − Z_{1}, g_{3}, g_{4}}
∪ {2Z_{1}Z_{3} − 2Z_{3} + 3Z_{2}^{2} + 6Z_{1}Z_{2} − 9Z_{2} − 2Z_{1} + 2}
∪ {2Z_{2}Z_{3} − 2Z_{3} + 3Z_{2}^{2} − 4Z_{1}Z_{2} − 5Z_{2} + 4Z_{1} + 2}
h_{1} := 2Z_{3}^{2} − 8Z_{3} + 15Z_{2}^{2} + 30Z_{1}Z_{2} − 45Z_{2} + 6
H_{2} := {Z_{1}^{2} − Z_{1}, Z_{1}Z_{2}}
h_{2} := Z_{2}^{2} − 2Z_{2},
h_{3} := 2Z_{3} + 3Z_{2} − 4Z_{1} − 2,
H_{3} := {Z_{1}^{2} − Z_{1}, Z_{1}Z_{2}}
h_{4} := Z_{2}^{2} − Z_{2}
h_{5} := Z_{3} + 3Z_{2} − 2Z_{1} − 1

we obtain

J_{1} := {g_{1}, …, g_{8}}

J_{2} := 𝕀({Lp(g_{i}), 1 ≤ i ≤ 7}) = 𝕀(Z_{1} − 2, Z_{2})

𝔱_{1} := {Z_{1} − 2, Z_{2}} Solve(J_{2}) := {𝔱_{1}} 𝔱_{1} := {Z_{1} − 2, Z_{2}, Z_{3}^{3} − 3Z_{3}^{2} + 2Z_{3}} J_{3} := J_{1} : (Z_{1} − 2) = 𝕀(H_{1} ∪ {h_{1}})

J_{5} := 𝕀({Lp(h) : h ∈ H_{1}}) = 𝕀(Z_{1} − 1, Z_{2} − 1)

𝔱_{2} := 𝕀({Z_{1} − 1, Z_{2} − 1}) Solve(J_{5}) := {𝔱_{2}} 𝔱_{2} := {Z_{1} − 1, Z_{2} − 1, Z_{3}^{2} − 4Z_{3} + 3} J_{6} := J_{3} : (Z_{2} − 1) = 𝕀(H_{2} ∪ {h_{3}, h_{2}})

J_{7} := 𝕀({Lp(h) : h ∈ H_{2}}) = 𝕀(Z_{1}, Z_{1}^{2} − Z_{1}) = 𝕀(Z_{1}) 𝔱_{3} := {Z_{1}, h_{2}, h_{3}} = {Z_{1}, Z_{2}^{2} − 2Z_{2}, 2Z_{3} + 3Z_{2} − 2}

J_{8} := J_{6} : Z_{1} = 𝕀(Z_{1} − 1, Z_{2})
𝔱_{4} := {Z_{1} − 1, Z_{2}, h_{3}} = {Z_{1} − 1, Z_{2}, 2Z_{3} − 6}

Solve(J_{6}) := {𝔱_{3}, 𝔱_{4}} J_{9} := (J_{3} + 𝕀(Z_{2} − 1)) : (Z_{1} − 1) := 𝕀(Z_{1}, Z_{2} − 1, Z_{3} + 2)

𝔱_{5} := {Z_{1}, Z_{2} − 1, Z_{3} + 2} Solve(J_{9}) := {𝔱_{5}}

Solve(J_{3}) := {𝔱_{i}, 2 ≤ i ≤ 5} J_{4} := (J_{3} + 𝕀(Z_{2} − 1)) : (Z_{1} + Z_{2} − 2) = Q Solve(J_{4}) := ∅

Solve(J_{1}) := {𝔱_{i}, 1 ≤ i ≤ 5}

and

Z(J) = {b_{j} : 1 ≤ j ≤ 9} = ⨆_{i=1}^{5} Z(𝔱_{i})

with

Z(𝔱_{1}) = {b_{3}, b_{8}, b_{9}}  Z(𝔱_{2}) = {b_{6}, b_{7}}
Z(𝔱_{3}) = {b_{1}, b_{4}}  Z(𝔱_{4}) = {b_{5}} Z(𝔱_{5}) = {b_{2}}

�

42.9 Rouillier: Rational Univariate Representation

Let us assume we are given a zero-dimensional ideal J ⊂ Q via a Gröbner representation

b = {[b_{1}], …, [b_{s}]} ⊂ A = Q/J, A_{h} := (a_{ij}^{(h)}) = M([Z_{h}], b), 1 ≤ h ≤ r

and let us remark that, via a direct application of Alonso–Raimondo–Traverso Algorithm (Remark 40.8.1) we can reduce ourselves with good complexity to the case in which

(1) J is radical,

(2) we have a linear form Y := ∑_{h} c_{h}Z_{h} which is a separating element of Z(J).

where we recall that

Definition 42.9.1. A polynomial f ∈ Q is called a separating element of

Z(J) iff, for each α, β ∈ Z(J), we have α ≠ β ⟹ f(α) ≠ f(β).  �

The application of Alonso–Raimondo–Traverso Algorithm has the further advantage that the obtained separating linear form Y := ∑_{h} c_{h}Z_{h} is an allgemeine coordinate for J so that (Corollary 34.3.4) Alonso–Raimondo– Traverso Algorithm returns a triangular set

(g_{0}(Y), Z_{1} − g_{1}(Y), …, Z_{r} − g_{r}(Y) ⊂ K[Y, Z_{1}, …, Z_{r}],  (42.1)

of the ideal J^{+} := J + (Y − ∑_{h} c_{h}Z_{h}) ⊂ K[Y, Z_{1}, …, Z_{r}], where g_{i} ∈ K[Y], deg(g_{i}) < deg(g_{0}) = #(Z(J)) and (J being radical) g_{0} is squarefree.

Example 42.9.2. For the radical ideal J ⊂ ℂ[Z_{1}, Z_{2}, Z_{3}] discussed in Examples 39.2.3 and 40.3.2 and the separating element/allgemeine coordinate Y = −3Z_{1} + Z_{2} + 3Z_{3} we have

g_{0} = Y^{9} + Y^{8} − 90Y^{7} − 142Y^{6} + 2489Y^{5}
+ 4689Y^{4} − 20880Y^{3} − 31428Y^{2} + 45360Y,
389188800g_{1} = −8611Y^{8} + 29288Y^{7} + 697698Y^{6}
− 2278040Y^{5} − 15347699Y^{4} + 56296512Y^{3}
+ 44649972Y^{2} − 473227920Y + 778377600,
640640g_{2} = 19Y^{8} − 108Y^{7} − 1426Y^{6} + 7808Y^{5}

+ 31851Y^{4} − 167652Y^{3} − 185004Y^{2} + 955152Y,
778377600g_{3} = −24917Y^{8} + 102316Y^{7} + 1972926Y^{6}
− 7718320Y^{5} − 43595053Y^{4} + 180492084Y^{3}
+ 164226564Y^{2} − 1073833200Y + 1556755200.

�

Proposition 42.9.3 (Alonso–Becker–Roy–Wörmann). With the current notation and setting

Z(J) := {α_{1}, …, α_{s}} ⊂ K^{r}, α_{i} = (a_{1}^{(i)}, …, a_{r}^{(i)}), β_{i} := ∑_{h} c_{h}a_{h}^{(i)}

there are polynomials h_{1}(Y), …, h_{r}(Y) ∈ K[Y], deg(h_{i}) < deg(g_{0}), such that

J^{+} = 𝕀 (g_{0}(Y), g′_{0}(Y)Z_{1} − h_{1}(Y), …, g′_{0}(Y)Z_{r} − h_{r}(Y)) ⊂ K[Y, Z_{1}, …, Z_{r}]. (42.2)

Moreover, for each ι, 1 ≤ ι ≤ r, we have

h_{ι}(Y) = ∑_{i=1}^{s} a_{ι}^{(i)} ∏_{j≠i} (Y − β_{j}) .  (42.3)

Proof. g_{0}(Y) being squarefree, g′_{0}(Y) is invertible in K[Y]/g_{0}; thus h_{i} := Rem(g′_{0}g_{i}, g_{0}) satisfy the required property.

Since g′_{0}(Y) = ∑_{i=1}^{s} ∏_{j≠i} (Y − β_{j}), for each l, 1 ≤ l ≤ s we have

g′_{0}(β_{l})a_{ι}^{(l)} = a_{ι}^{(l)} ∑_{i=1}^{s} ∏_{j≠i} (β_{l} − β_{j}))
= a_{ι}^{(l)} ∏_{j≠l} (β_{l} − β_{j})

= ∑_{i=1}^{s} a_{ι}^{(i)} ∏_{j≠i} (β_{l} − β_{j})
= h_{ι}(β_{l}).

�

Remark 42.9.4. Compare Proposition 42.9.3 with Kronecker’s result (41.3); the only difference is that here the assumption of the primality of the ideal J

is relaxed to radicality.  �

Remark 42.9.5 (Alonso–Becker–Roy–Wörmann). Denoting S the size of the elements a_{ij}^{(ι)} in the matrices A_{ι}, clearly (42.3) grants that the coefficients in the Kronecker parametrization (42.2) have size O(Ss) giving a strong advantage with the O(Ss^{2}) size of the coefficients of the allgemeine basis

(42.1).  �

Example 42.9.6. In the setting discussed in Examples 42.9.2 we have

g′_{0}(Y) = 9Y^{8} + 8Y^{7} − 630Y^{6} − 852Y^{5} + 12445Y^{4}
+ 18756Y^{3} − 62640Y^{2} − 62856Y + 45360,
h_{1}(Y) = 9Y^{8} + 5Y^{7} − 638Y^{6} − 668Y^{5} + 13655Y^{4}
+ 15591Y^{3} − 92178Y^{2} − 76896Y + 90720,
h_{2}(Y) = 5Y^{8} − 348Y^{6} − 62Y^{5} + 7155Y^{4} + 2790Y^{3} − 39852Y^{2} − 20088Y,
h_{3}(Y) = 7Y^{8} + 65Y^{7} − 380Y^{6} − 3966Y^{5} + 3455Y^{4}
+ 56421Y^{3} − 5562Y^{2} − 191160Y + 90720.

Corollary 42.9.7 (Alonso–Becker–Roy–Wörmann). For each f ∈ Q, there is h_{f}(Y) ∈ K[Y] such that

g′_{0}(Y)f(Z_{1}, …, Z_{r}) − h_{f}(Y) ∈ J^{+}, deg(h_{f}) < deg(g_{0}).

Proof. It is sufficient to set h_{f}(Y) := Rem(f(h_{1}(Y), …, h_{r}(Y)), g_{0}(Y)). �

Both Proposition 42.9.3 and Corollary 42.9.7 are existential results; thus we need a computational definition of h_{f}(Y); to obtain it we consider a new variable S, the extension field K(S), the ideal

J^{e} := JK(S)[Z_{1}, …, Z_{r}] ⊂ K(S)[Z_{1}, …, Z_{r}] = Q ⊗_{K} K(S),

the algebra Ā := K(S)[Z_{1}, …, Z_{r}]/J^{e} = A ⊗_{K} K(S), the element f̄:= Y + Sf ∈ Ā, the matrix Af̄.

Theorem 42.9.8 (Alonso–Becker–Roy–Wörmann). With the current notation, it holds:

(1) the minimal polynomial mf̄(T) ∈ K(S)[T] of Af̄ is

mf̄(T) = ∏_{i=1}^{s} (T − β_{i} − Sf(α_{i})) ∈ K[S, T];

(2) denoting p(S, T) := \frac{∂mf̄}{∂S}, we have

p(0, T) = − ∑_{i=1}^{s} f(α_{i}) ∏_{{j=1 \atop j≠i}}^{r}(T − β_{j});

(3) for each α ∈ Z(J) it holds f(α) = \frac{p(0,α)}{g′_{0}(α)};

(4) h_{f}(T) = p(0, T).

Proof. (1) is obvious, (2) requires a trivial verification and (4) is a direct consequence of (3); so we have just to prove (3): for each ι, 1 ≤ ι ≤ s we have

p(0, α_{ι}) = − ∑_{i=1}^{s} f(α_{i}) ∏_{{j=1 \atop j≠i}}^{s}(β_{ι} − β_{j})

= −f(α_{ι}) ∏_{{j=1 \atop j≠ι}}^{r}(β_{ι} − β_{j})

= −f(α_{ι})g′_{0}(α_{ι}).

�

Remark 42.9.9 (Alonso–Becker–Roy–Wörmann). The computation of the Kronecker parametrization (42.2) does not require to assume that J is radical; assuming as known s = #Z(J) (an efficient way for computing it is discussed in Corollary 42.9.13 below) and denoting χ(T) and m(T) respectively the characteristic and the minimal polynimials of A_{Y} we have

g_{0}(T) = \sqrt{χ(T)} = \sqrt{m(T)}.

Once a linear form Y is fixed, the minimal polynomial m(T) of A_{Y} which coincides with the minimal polynomial of the element Y ∈ A can be directly obtained by checking the successive powers [1], [Y], [Y^{2}], … for linear dependency; Y is then a separating element if and only if deg(\sqrt{m}) = #Z(J).

In order to make this approach effictive, all one needs is the availability of a finite set of linear forms which is granted to contain at least a separating element. Such a finite set is provided by Chistov–Grigoriev Corollary 35.6.4.

�

Remark 42.9.10. Alternatively, once #Z(J) is known, a separating linear form Y can be obtained by an easy adaptation of Alonso–Raimondo Algorithm 35.7.1^{56}aimed to avoid the evaluation of the swelling coefficients of the g_{i}s, i > 0:

(1) by linear algebra on the Gröbner descriptions of [1], [Y], [Y^{2}], … compute the minimal polynomial m[Y] ∈ K[Y] such that m(Y) ∈ J^{+}.

(2) if d := deg(\sqrt{m}) < #Z(J) then set j = 1 and

(a) while j ≤ r, verify, whether [Z_{j}], [1], [Y], [Y^{2}], …, [Y^{d−1}] are linearly dependent;

(b) if so set j := j + 1 and go to (2.a);

(c) if instead they are linearly dependent set Y := Y + cZ_{j} and go to (1)

(3) if deg(\sqrt{m}) = #Z(J), then • J^{+} + (\sqrt{m}) is radical,

• Y is a separating linear form and

56 Compare also Remark 40.8.1.

• \sqrt{m} is its minimal polynomial. �

Lemma 42.9.11. Let f ∈ Q be a separating element of Z(J) and let s = #Z(J). Then {1, [f], [f^{2}], …, [f^{s−1}]} is a K-linearly independent set of A.

Proof. Assume that m(T) := ∑_{i=0}^{s−1} a_{i}T^{i} ∈ k[Y] is such that m(f) ≡ 0 mod J; since f a separating element of Z(J), the polynomial m(T) has the s distinct

roots {f(α), α ∈ Z(J)} giving a contradiction.  �

For every polynomial h ∈ Q, we can consider the bilinear map

ℓ_{h} : A × A → K, (p, q) ↦ Tr(A_{hpq})

and the corresponding quadratic form associated to ℓ_{h}

Q_{h}(x_{1}, …, x_{s}) := ∑_{j,l} γ_{j,l}^{(h)}x_{j}x_{l}

which satisfies ℓ_{h}(p, p) = Tr(A_{hp^{2}}) = ∑_{j,l} γ_{j,l}^{(h)}c_{j}c_{l} = Q_{h}(c_{1}, …, c_{s}) for each p = ∑_{i=1}^{s} c_{i}[b_{i}] ∈ A = Span_{K}{[b_{1}], …, [b_{s}]}.

Proposition 42.9.12 (Rouillier). For a polynomial h ∈ Q, the quadratic form Q_{h} has #{α ∈ Z(J) : h(α) ≠ 0} as rank.

Proof. Let f ∈ Q be a separating element of Z(J). By the Lemma above, the set {1, [f], [f^{2}], …, [f^{s−1}]} is K-linearly independent and thus can be completed to a basis

{1, [f], [f^{2}], …, [f^{s−1}], [b_{s+1}], …, [b_{s}]} = {[b_{1}], …, [b_{s}]}

of A. For each p ∈ Q let c_{j} ∈ K be such that

[p] = ∑_{j=0}^{s−1} c_{j}[f]^{j} + ∑_{j=s+1}^{s} c_{j}[b_{j}].

Thus, setting Y_{i} := ∑_{j=1}^{s} c_{j}b_{j}(α_{i}), 1 ≤ i ≤ s, by Corollary 40.5.2 we have

Q_{h} = Tr(A_{hp^{2}}) = ∑_{i=1}^{s} s_{i}h(α_{i}) (∑_{j=1}^{s} c_{j}b_{j}(α_{i}))^{2} = ∑_{i=1}^{s} s_{i}h(α_{i})Y_{i}^{2}.

The matrix

( \begin{matrix}1 & f(α_{1}) & f(α_{1})^{2} & ⋯ & f(α_{1})^{s−1} \\ 1 & f(α_{2}) & f(α_{2})^{2} & ⋯ & f(α_{2})^{s−1} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 1 & f(α_{s}) & f(α_{s})^{2} & ⋯ & f(α_{s})^{s−1} \end{matrix} )

is invertible — being Vandermonde since f ∈ Q separates Z(J), — and a submatrix of the one associated to the linear forms that define the linear change of variables Y_{i}.

Thus the Y_{i}s are linearly independent and (compare Theorem 13.5.2) the rank of Q_{h} is the number of roots of J which are not roots of h.

Corollary 42.9.13. The rank of

Q_{1}(x_{1}, …, x_{s}) := ∑_{j,l} γ_{j,l}^{(1)}x_{j}x_{l} = ∑_{j,l} Tr(A_{b_{j}b_{l}})x_{j}x_{l}

is s = #Z(J).  �

Once a Kronecker parametrization (42.2) is obtained via Theorem 42.9.8, the multiplicity of each root can be obtained in the following way: denoting, for each f ∈ Q, B_{f} the matrix representing the endomorphism

Φ̄ : Q/\sqrt{J} → Q/\sqrt{J}, Φ̄([g]) ↦ [fg]

we have

Lemma 42.9.14 (Alonso–Becker–Roy–Wörmann). Let Y be a a separating linear form of Z(J); then

(1) The matrix ( \begin{matrix}Tr(B_{1}) & ⋯ & Tr(B_{Y^{s−1}}) \\ ⋮ & ... & ⋮ \\ Tr(B_{Y^{s−1}}) & ⋯ & Tr(B_{Y^{2s−2}}) \end{matrix} ) , is invertible;

(2) let a_{0}, …, a_{s−1} ∈ K be the unique solution of the linear system

( \begin{matrix}Tr(B_{1}) & ⋯ & Tr(B_{Y^{s−1}}) \\ ⋮ & ... & ⋮ \\ Tr(B_{Y^{s−1}}) & ⋯ & Tr(B_{Y^{2s−2}}) \end{matrix} ) · ( a_{0} ; ⋮ ; a_{s−1} ) = ( Tr(A_{1}) ; ⋮ ; Tr(A_{Y^{s−1}}) )

and let F (Y) := ∑_{l=0}^{s−1} a_{l}Y^{l}; then

F (α_{i}) = s_{i} = mult(α_{i}, J), ∀α_{i} ∈ Z(J).

Proof. (1) holds since the matrix is Hankel.

Ad(2): we have, for each i, 1 ≤ i ≤ s

∑_{i=1}^{s} s_{i}β_{i}^{j} = Tr(A_{Y^{j}})

= ∑_{l=0}^{s−1} a_{l} Tr(B_{Y^{j+l}})

= ∑_{l=0}^{s−1} a_{l} ∑_{i=1}^{s} β_{i}^{j+l}

= ∑_{i=1}^{s} (∑_{l=0}^{s−1} a_{l}β_{i}^{l}) β_{i}^{l}

= ∑_{i=1}^{s} F (α_{i})β_{i}^{j};

since the matrix (β_{i}^{j}) is Vandermonde, we have F (α_{i}) = s_{i} for each i. �

Up to now, in order to represent the roots of I we alternatively,

• assumed J to be radical in order to apply Corollary 34.3.4 or

• applied Proposition 42.9.3, using the minimal polynomial m(Y) of Y ∈ A.

In both cases, we lose the multiplicity of each root, which we recover via Lemma 42.9.14.

An improvement allows to remove the requirement that J be radical and to directly use the characteristic polynomial χ(T) = ∏_{i=1}^{s}(T − β_{i})^{s_{i}}, thus directly deducing multiplicities.

Proposition 42.9.15 (Rouiller). Let J ⊂ Q be a zero-dimensional ideal, not necessarily radical and set

Z(J) := {α_{1}, …, α_{s}} ⊂ K^{r}, α_{i} = (a_{1}^{(i)}, …, a_{r}^{(i)}), s_{i} := mult(α_{i}, J).

Let f ∈ Q be a separating element of Z(J); denote χ := χ_{f} := ∑_{i=0}^{s} c_{i}T^{s−i} the charactristic polynomial of A_{f} and β_{i} := f(α_{i}).

For each h ∈ Q denote γ_{h}(T) := ∑_{i=1}^{s} s_{i}h(α_{i}) ∏_{{j=1 \atop j≠i}}^{s}(T − β_{j}).

Then:

(1) χ = χ_{f} = ∏_{i=1}^{s} (T − β_{i}))^{s_{i}}.

(2) \frac{χ′(T)}{χ(T)} = ∑_{j≥0} \frac{Tr(A_{f^{j}})}{T^{j+1}};

(3) (s − i)c_{i} = ∑_{j=0}^{i} c_{i−j} Tr(A_{f^{j}}) for i = 0, .., s;

(4) γ_{h}(β_{l}) = s_{l}h(α_{l}) ∏_{{j=1 \atop j≠l}}^{s}(β_{l} − β_{j});

(5) h(α_{l}) = \frac{γ_{h}(β_{l})}{γ_{1}(β_{l})} for 1 ≤ l ≤ s;

(6) γ_{h}(T) = \sqrt{χ(T)} ∑_{j≥0} \frac{Tr(A_{hf^{j}})}{T^{j+1}};

(7) setting \sqrt{χ} := ∑_{i=0}^{s} a_{i}T^{s−i}, it holds

γ_{h}(T) = ∑_{i=0}^{s−1} ∑_{j=0}^{s−i−1} Tr(A_{hf^{j}})a_{i}T^{s−i−j−1};

(8) γ_{1}(T) = \frac{χ′(T)}{gcd(χ(T),χ′(T))};

(9) gcd(γ_{1}(T), χ(T)) = 1; (10) for each i, s_{i} = \frac{γ_{1}(β_{i})}{(\sqrt{χ})′(β_{i})};

(11) the squarefree decomposition (Definition 4.7.2) χ = ∏_{l} χ_{l}^{l} of χ is obtained via χ_{l} := gcd(γ_{1}(T) − l(\sqrt{χ})′(T), \sqrt{χ}).

Proof.

(1) See Corollary 40.5.2.

(2) We have

\frac{χ′(T)}{χ(T)} = ∑_{i=1}^{s} \frac{s_{i}}{T − β_{i}}

= ∑_{i=1}^{s} \frac{s_{i}}{T} \frac{1}{1 − \frac{β_{i}}{T}}

= ∑_{i=1}^{s} \frac{s_{i}}{T} ∑_{j≥0} (\frac{β_{i}}{T} )^{j}

= ∑_{j≥0} \frac{∑_{i=1}^{s} s_{i}β_{i}^{j}}{T^{j+1}}

= ∑_{j≥0} \frac{Tr(A_{f^{j}})}{T^{j+1}}

(3) We have

∑_{i=0}^{s}(s − i)c_{i}T^{s−i−1} = χ′(T)

= χ(T) ∑_{j≥0} \frac{Tr(A_{f^{j}})}{T^{j+1}}

= ∑_{l=0}^{s−1} ∑_{j=0}^{s−l−1} c_{l} Tr(A_{f^{j}})T^{s−l−j−1}

= ∑_{i=0}^{s−1} ∑_{j=0}^{i} c_{i−j} Tr(A_{f^{j}})T^{s−i−1}.

(4) We have γ_{h}(β_{l}) = ∑_{i=1}^{s} s_{i}h(α_{i}) ∏_{{j=1 \atop j≠i}}^{s}(β_{l} − β_{j}) = s_{l}h(α_{l}) ∏_{{j=1 \atop j≠l}}^{s}(β_{l} − β_{j}).

(5) Obvious.

(6) We have

\frac{γ_{h}(T)}{\sqrt{χ(T)}} = ∑_{i=1}^{s} \frac{s_{i}h(α_{i})}{T − β_{i}} = ∑_{j≥0} \frac{∑_{i=1}^{s} s_{i}h(α_{i})β_{i}^{j}}{T^{j+1}} = ∑_{j≥0} \frac{Tr(A_{hf^{j}})}{T^{j+1}}.

(7) A direct consequence of (6).

(8) Obvious.

(9) Obvious.

(10) \frac{γ_{1}(T)}{(\sqrt{χ})′(T)} = \frac{∑_{i=1}^{s} s_{i} ∏_{{j=1 \atop j≠i}}^{s} (T −β_{j})}{∑_{i=1}^{s} ∏_{{j=1 \atop j≠i}}^{s} (T −β_{j})} whence the claim.

(11) The claim follows easily from

γ_{1}(T) − l(\sqrt{χ})′(T) = ∑_{i=1}^{s}(s_{i} − l) ∏_{{j=1 \atop j≠i}}^{s}(T − β_{j})

= ∑_{{i=1 \atop s_{i}≠l}}^{s}(s_{i} − l) ∏_{{j=1 \atop j≠i}}^{s}(T − β_{j})

= ( ∏_{{i=1 \atop s_{i}=l}}^{s}(T − β_{i})) (∑_{{i=1 \atop s_{i}≠l}}^{s}(s_{i} − l) ∏_{{j=1 \atop j≠i,s_{j} ≠l}}^{s}(T − β_{j}))

�

Definition 42.9.16 (Rouillier). Let J ⊂ Q be a zero-dimensional ideal. A Univariate Representation (χ, Φ) of J is the assignement of polynomials χ(T), γ_{0}(T), γ_{1}(T), …, γ_{r}(T) ∈ K[T] which defines a K-isomorphism

Φ : \{α ∈ K : χ(α) = 0\} → Z(J) : α ↦ (\frac{γ_{1}(α)}{γ_{0}(α)}, …, \frac{γ_{r}(α)}{γ_{0}(α)})

which satisfies mult(Φ(α), J) = mult(α, 𝕀(χ)).

If moreover f ∈ Q is a separating element of Z(J) and χ := χ_{f} is the charactristic polynomial of A_{f}, the univariate representation (χ, Φ) is called

the Rational Univariate Representation (RUR) of J associated to f.  �

Corollary 42.9.17 (Rouillier). With the assumptions and notations of Proposition 42.9.15 and setting

Φ : \{f(α), α ∈ Z(J)\} → Z(J) : β = f(α) ↦ (\frac{γ_{Z_{r}}(β)}{γ_{1}(β)}, …, \frac{γ_{Z_{1}}(β)}{γ_{1}(β)}) = α

(χ_{f}, Φ) is the Rational Univariate Representation of J associated to f. �

Remark 42.9.18 (Rouillier). For a Rational Univariate Representation

(χ, Φ), Φ : \{α ∈ K : χ(α) = 0\} → Z(J) : α ↦ (\frac{γ_{1}(α)}{γ_{0}(α)}, …, \frac{γ_{r}(α)}{γ_{0}(α)})

of J and a factorization χ(T) = ∏_{i} χ_{i}(T), gcd(χ_{i}, χ_{j}) = 1 setting

γ_{ji}(T) := Rem(γ_{j}(T), χ_{i}) for each i, j

and

Φ_{i} : \{α ∈ K : χ_{i}(α) = 0\} → \{(\frac{γ_{1i}(α)}{γ_{0i}(α)}, …, \frac{γ_{ni}(α)}{γ_{0i}(α)})\}

we have that (χ_{i}, Φ_{i}) is a Rational Univariate Representation, for each i, and

it holds Z(J) = ⊔_{i} \{(\frac{γ_{1i}(α)}{γ_{0i}(α)}, …, \frac{γ_{ni}(α)}{γ_{0i}(α)}) : α ∈ K, χ_{i}(α) = 0\} .  �

Remark 42.9.19. If J is radical, the representation of Z(J) proposed in Propo-

sition 42.9.3 is a RUR of J associated to ∑_{h} c_{h}Z_{h}.  �

Algorithm 42.9.20 (Rouillier). Given a zero-dimensional ideal J ⊂ Q via a Gröbner representation

b = {[b_{1}], …, [b_{s}]} ⊂ A = Q/J, A_{h} := (a_{ij}^{(h)}) = M([Z_{h}], b), 1 ≤ h ≤ r

and assuming to have the matrices A_{b_{i}} representing the endomorphisms Φ_{b_{i}} : A → A, a RUR of J associated to a linear form can therefore be computed by the following procedure:

(1) For each j, l, 1 ≤ i ≤ s compute Tr(A_{b_{j}b_{l}});

(2) Compute s := #Z(J) as rank of Q_{1}(x_{1}, …, x_{s}) := ∑_{j,l} Tr(A_{b_{j}b_{l}})x_{j}x_{l} (Corollary 42.3.6);

(3) Repeatedly, choose (via Corollary 35.6.4) a linear form Y := ∑_{h} c_{h}Z_{h} and compute, via Proposition 42.9.15(3), the characterisitc polynomial χ_{Y} of A_{Y} until deg(\sqrt{χ_{Y}}) = #Z(J) thus granting that Y is a separating element. Alternatively deduce via Remark 42.9.10 a separating linear form Y := ∑_{h} c_{h}Z_{h} and compute the characteristic polynomial χ_{Y} of A_{Y} via Proposition 42.9.15(3).

(4) Compute γ_{1}(T), γ_{Z_{1}}(T), …, γ_{Z_{r}}(T) via Proposition 42.9.15(6-7).

It can be proved that its complexity is in O(s^{3} + rs^{2}) arithmetic operations in K and that, in the case K = ℚ, the cost is in O((s^{3} + rs^{2})M(ls^{2})) binary arithmetic operations, where l denotes the bit-size of the entries of the matrices A_{h} and M(ℓ) denotes the cost of multiplying two integers of bit-size

ℓ.  �

Remark 42.9.21 (Dahen). Clearly (Compare Proposition 42.9.3) the relation between the elements of the Allegemiane basis (42.1) and the RUR (42.2) is given by h_{i} := Rem(g′_{0}g_{i}, g_{0}).

The better behavieour of Kronecker’s parametrization w.r.t. Allegemiane bases can be generalized to triangular sets.

In fact, given a triangularizable zero-dimensional ideal J ⊂ ℚ[Z_{1}, …, Z_{r}] via a set G := (g_{1}, …, g_{m}) ⊂ ℚ[Z_{1}, …, Z_{r}], denoting

T := (f_{1}, …, f_{r}) ⊂ ℚ[Z_{1}, …, Z_{r}]

its triangular set and, for each i, 1 ≤ i < r,

N_{i+1} := NF (f_{i+1} ∏_{j=1}^{i} \frac{∂f_{j}}{∂Z_{j}} , 𝕀(f_{1}, …, f_{i}))

the normal form of f_{i+1} ∏_{j=1}^{i} \frac{∂f_{j}}{∂Z_{j}} w.r.t. the Gröbner basis (f_{1}, …, f_{i}), both theoretical and pratical analysis suggest that the height^{57} of the triangular

set N := (f_{1}, N_{2}, …, N_{r}) of J is better than the one of T^{58}.  �

57 for a set G := (g_{1}, …, g_{m}) ⊂ ℚ[Z_{1}, …, Z_{r}] its height is the value h(G) :=

max (log(c(g_{i}, τ) : 1 ≤ i ≤ r, τ ∈ W) . 58 We have h(T) = O(rhd^{2r}) and h(N) = O(rhd^{r}) where d = max(deg(g_{i})) and

h = h(G).

192 42. Lazard II

43. Lagrange II

Given a (squarefree) polynomial f ∈ k[T], a natural question^{1} is to determine its Galois group (Definition 14.1.3) over k. Based on classical techniques as the representation of a group of finite order as a permutation group (Section 43.1) and as a permutation group of the set of roots of a separable polynomial (Section 43.2), Lagrange resolvents (Section 43.3 and 43.5) and Cachy modules (Section 43.4) recently the problem has been completely solved (Section 43.6 and 43.7) by Annick Valibouze and Jean-Marie Arnaudies for polynomials f of degree bounded by 11.

43.1 Representation of Groups as Permutation Groups

Let^{2} G be a finite group.

Example 43.1.1. Throughout this section, as an example we take as G the alternative group A_{4} whose 12 elements we denote

\begin{aligned}g_{1} & = & Id, & g_{2} & = & (1, 3, 2), & g_{3} & = & (1, 2), (3, 4), \\ g_{4} & = & (1, 2, 3), & g_{5} & = & (2, 4, 3), & g_{6} & = & (2, 3, 4), \\ g_{7} & = & (1, 4, 3), & g_{8} & = & (1, 4, 2), & g_{9} & = & (1, 3, 4), \\ g_{10} & = & (1, 3), (2, 4), & g_{11} & = & (1, 2, 4), & g_{12} & = & (1, 4), (2, 3) \end{aligned}

and whose corresponding multiplication table is

1 Other equally natural questions are

• to solve f by radicals, in case it is possible, and

• the Galois inverse problem of determining, given a finite group G, a polynomial f ∈ k[T] for which G is its Galois group. Such questions are just mentioned here but not discussed in this book.

2 For this theory, compare Burnside W., Theory of groups of finite order, Cambridge Universiy Press (1911) Ch. XII

1 2 3 4 5 6 7 8 9 10 11 12
1 1 2 3 4 5 6 7 8 9 10 11 12
2 2 4 7 1 3 8 5 10 12 6 9 11
3 3 6 1 9 8 2 11 5 4 12 7 10
4 4 1 5 2 7 10 3 6 11 8 12 9
5 5 10 4 11 6 1 12 7 2 9 3 8
6 6 9 11 3 1 5 8 12 10 2 4 7
7 7 8 2 12 10 4 9 3 1 11 5 6
8 8 12 9 7 2 3 10 11 6 4 1 5
9 9 3 8 6 11 12 1 2 7 5 10 4
10 10 11 12 5 4 7 6 9 8 1 2 3
11 11 5 6 10 12 9 4 1 3 7 8 2
12 12 7 10 8 9 11 2 4 5 3 6 1

�

Impose on the set of the subgroups H ⊂ G the relation

H_{1} ∼ H_{2} ⟺ exists τ ∈ G : H_{1} = τ^{−1}H_{2}τ

and denotes E := {C_{1}, …, C_{s}}, the set of all the conjugacy classes. We associate to each such class its degree

deg(C_{i}) := [G : H], H ∈ C_{i},

and its weight

w(C_{i}) := #G/ deg(C_{i}) = #H, H ∈ C_{i},

we enumerate E so that^{3}

w(C_{1}) ≤ w(C_{2}) ≤ ⋯ ≤ w(C_{s})

and we impose a partial ordering ⪯ on E setting C_{i} ⪯ C_{j} if the following equivalent conditions

• there are H ∈ C_{i} and H′ ∈ C_{j} such that H ⊂ H′,

• for each H ∈ C_{i} there is H′ ∈ C_{j} such that H ⊂ H′,

hold.

Since in this setting we have [G : H′] = [G : H][H : H′] we also have C_{i} ⪯ C_{j} ⟹ i ≤ j.

Example 43.1.2. G = A_{4} has five conjugacy classes, each consisting of the subgroups of order (respectively) 1, 2, 3, 4, 12:

C_{1} has deg(C_{1}) = 12, w(C_{1}) = 1 and consists of

H_{11} := H_{1} := {Id};

3 In particular: C_{1} = {{Id_{G}}} and C_{s} = {G}.

C_{2} has deg(C_{2}) = 6, w(C_{2}) = 2 and contains of

\begin{aligned}H_{21} & := & H_{2} & := & \{Id, g_{3}\}, \\ H_{22} & := & g_{4}H_{2}g_{4}^{−1} & = & \{Id, g_{10}\}, \\ H_{23} : & := & g_{2}H_{2}g_{2}^{−1} & = & \{Id, g_{12}\}; \end{aligned}

C_{3} has deg(C_{3}) = 4, w(C_{3}) = 3 and contains of

H_{31} := H_{3}  := {Id, g_{2}, g_{4}},
H_{32} := g_{7}H_{3}g_{7}^{−1} = {Id, g_{5}, g_{6}},
H_{33} := g_{5}H_{3}g_{5}^{−1} = {Id, g_{7}, g_{9}},
H_{34} := g_{6}H_{3}g_{6}^{−1} = {Id, g_{8}, g_{11}};

C_{4} has deg(C_{4}) = 3, w(C_{4}) = 4 and contains of

H_{41} := H_{4} := {Id, g_{3}, g_{10}, g_{12}};

C_{5} has deg(C_{5}) = 1, w(C_{5}) = 12 and consists of

H_{51} := H_{5} := G.

Naturally we have

C_{1} ≺ C_{2} ≺ C_{4} ≺ C_{5}, C_{1} ≺ C_{3} ≺ C_{5}.

�

Let now E be a finite set, n := #E, and denote S_{E} the group of the permutations of the set E.

Definition 43.1.3. Each group morphism Φ : G → S_{E} is called a representation of G as a permutation group of degree n.

It is said to be

• faithful if ker(Φ) = {Id_{G}},

• transitive if for each x, x_{0} ∈ E there is g ∈ G : Φ(g)(x_{0}) = x.

A representation Ψ : G → S_{F} is called equivalent to Φ (denoted: ψ ∼ Φ) if there is a bijection Θ : E → F satisfying Ψ(g) = Θ ∘ Φ(g) ∘ Θ^{−1} for each g ∈ G :

F ⟼^{Ψ(g)} F
↑Θ  ↑Θ
E ⟼^{Φ(g)} E

�

Remark that

(1) ker(Φ) = ⋂_{x∈E}{g ∈ G : Φ(g)(x) = x}; (2) if Ψ is equivalent to Φ we have

• ker(Φ) = ker(Ψ),

• {{g ∈ G : Φ(g)(x) = x} : x ∈ E} = {{g ∈ G : Ψ(g)(x) = x} : x ∈ F };

(3) if Φ is transitive, then the set

C_{Φ} := {{g ∈ G : Φ(g)(x) = x} : x ∈ E} ∈ E

is a conjugacy class which is called the conjugacy class associated to Φ.

Assuming Φ to be transitive, fixing x_{0} ∈ E, and denoting

• H_{0} := {g ∈ G : Φ(g)(x_{0}) = x_{0}},

• (G/H_{0})_{l} := {gH_{0} : g ∈ G} the set of the left classes of H_{0},

• Θ the bijection Θ : E → (G/H_{0})_{l} defined Θ(x) = {g ∈ G : Φ(g)(x_{0}) = x},

• ρ : G → S_{(G/H_{0})_{l}} the representation ρ(h)(H′) = hH′, for each h ∈ G and each H′ ∈ (G/H_{0})_{l}

then

Lemma 43.1.4. ρ ∼ Φ.  �

Corollary 43.1.5. For a finite group G, denote 𝔈̄ the set of all transitive representations of G as permutation group and 𝔈 the set of the equivalency classes of the transitive representations of G as permutation group: 𝔈 := 𝔈̄/ ∼.

We obtain a bijection between 𝔈 and E by associating to each Γ ∈ 𝔈 the conjugacy class C_{Φ} associated to each Φ : G → S_{E} belonging to Γ.

In particular, there are transitive representations of G as permutation

group of degree n only if n = [G : H] for some subgroup H ⊂ G.  �

On the basis of this, we can associate to each C_{i} ∈ E a transitive representation ρ_{i} : G ↦ S_{d_{i}}, d_{i} = #C_{i} over the set of left classes of some H ∈ C_{i}.

Example 43.1.6. With G := A_{4} and H_{2} = {Id, g_{3}} we can consider the left classes of H_{2} which are

\begin{aligned}L_{21} & := & \{Id, g_{3}\}, & L_{22} & := & \{g_{2}, g_{7}\}, & L_{23} & := & \{g_{4}, g_{5}\}, \\ L_{24} & := & \{g_{6}, g_{11}\}, & L_{25} & := & \{g_{8}, g_{9}\}, & L_{26} & := & \{g_{10}, g_{12}\} \end{aligned}

thus obtaining a representation ρ_{2} : G → S_{6} in which we have

\begin{aligned}ρ_{2}(g_{1}) & = & Id, & ρ_{2}(g_{2}) & = & (1, 2, 3)(4, 5, 6), \\ ρ_{2}(g_{3}) & = & (2, 4)(3, 5) & ρ_{2}(g_{4}) & = & (1, 3, 2)(4, 6, 5), \\ ρ_{2}(g_{5}) & = & (1, 3, 4)(2, 6, 5), & ρ_{2}(g_{6}) & = & (1, 4, 3)(2, 5, 6), \\ ρ_{2}(g_{7}) & = & (1, 2, 5)(3, 6, 4), & ρ_{2}(g_{8}) & = & (1, 5, 4)(2, 6, 3), \\ ρ_{2}(g_{9}) & = & (1, 5, 2)(3, 4, 6), & ρ_{2}(g_{10}) & = & (1, 6)(2, 4), \\ ρ_{2}(g_{11}) & = & (1, 4, 5)(2, 3, 6), & ρ_{2}(g_{12}) & = & (1, 6)(3, 5). \end{aligned}

In the same way, with H_{3} = {Id, g_{2}, g_{4}}, the left classes of H_{3} are

\begin{aligned}L_{31} & := & \{Id, g_{2}, g_{4}\}, & L_{32} & := & \{g_{3}, g_{6}, g_{9}\} \\ L_{33} & := & \{g_{5}, g_{10}, g_{11}\}, & L_{34} & := & \{g_{7}, g_{8}, g_{12}\}, \end{aligned}

thus obtaining the representation ρ_{3} : G → S_{4} in which

\begin{aligned}ρ_{3}(g_{1}) & = & Id, & ρ_{3}(g_{2}) & = & (2, 4, 3), \\ ρ_{3}(g_{3}) & = & (1, 2)(3, 4), & ρ_{3}(g_{4}) & = & (2, 3, 4), \\ ρ_{3}(g_{5}) & = & (1, 3, 2), & ρ_{3}(g_{6}) & = & (1, 2, 3), \\ ρ_{3}(g_{7}) & = & (1, 4, 2), & ρ_{3}(g_{8}) & = & (1, 4, 3), \\ ρ_{3}(g_{9}) & = & (1, 2, 4), & ρ_{3}(g_{10}) & = & (1, 3)(2, 4), \\ ρ_{3}(g_{11}) & = & (1, 3, 4), & ρ_{3}(g_{12}) & = & (1, 4)(2, 3). \end{aligned}

Finally the left classes of H_{4} = {Id, g_{3}, g_{10}, g_{12}} are

L_{41} := {Id, g_{3}, g_{10}, g_{12}},
L_{42} := {g_{2}, g_{6}, g_{7}, g_{11}},
L_{43} := {g_{4}, g_{5}, g_{8}, g_{9}},

thus obtaining the representation ρ_{4} : G → S_{3} in which

\begin{aligned}ρ_{4}(g_{1}) & = & ρ_{4}(g_{3}) & = & ρ_{4}(g_{10}) & = & ρ_{4}(g_{12}) & = & Id, \\ ρ_{4}(g_{2}) & = & ρ_{4}(g_{6}) & = & ρ_{4}(g_{7}) & = & ρ_{4}(g_{11}) & = & (1, 2, 3), \\ ρ_{4}(g_{4}) & = & ρ_{4}(g_{5}) & = & ρ_{4}(g_{8}) & = & ρ_{4}(g_{9}) & = & (1, 3, 2). \end{aligned}

�

Let Φ : G → S_{E} be a (not necessarily transitive) representation of G as permutation group. Let C ∈ E be a conjugacy class and let H ⊂ G be any member of C; the number

m := #{e ∈ E : Φ(h)(e) = e : h ∈ H}

is clearly independent on the choice of H ∈ C but depends on the representation Φ of G as permutation group and on the conjugacy class C.

Definition 43.1.7. Such number m is called the mark of C in the representation Φ.

More in general if we consider the H-orbits of E and we denote α_{i} the number of orbits consisting of i elements^{4} such numbers are independent on the choise of H and depend only on Φ and C.

Remark that we have the relation #E = n = ∑_{i} iα_{i}.

4 So that, in particular m = α_{1}.

Example 43.1.8. Let us choose the representation ρ_{4} : G → S_{3} of G as permutation group and the group H := H_{2}.

Then we have 3 orbits of cardinality 1, so that m = 3.

If we instead choose ρ_{5} : G → S_{12} for H := H_{2} we obviously have 6 orbits

all of cardinality 2.  �

Setting C := C_{i} ⪯ C_{j} =: C′, using freely the corrent notation^{5} and defining

• B_{C′} := {C ∈ (G/H)_{l} : C ⊂ C′}, for each C′ ∈ (G/H′)_{l},

• B := {B_{C′} : C′ ∈ (G/H′)_{l}}

we have the partition (G/H)_{l} = ⊔_{B∈B}B in terms of the left-action of G. Thus we have a left-action of G on B which can be identified, via C′ ↦ B_{C′}, with the one on (G/H′)_{l}.

Let us fix, for any conjugacy class C_{j}, an associated transitive representation ρ_{j} : G → S_{C_{j}}, ρ_{j}(g) : H′ ↦ gH′.

By the consideration above on the action of G on B , the mark of C in the representation ρ_{j}^{6} is independent not only on the choice of H ∈ C_{i} but also on the choice of the transitive representation ρ_{j}. Such number therefore depends only on the couple (C, C′) = (C_{i}, C_{j}) and is called the incidence number or mark of (C_{i}, C_{j}) and will be denoted J(C_{i}, C_{j}) = m_{i}^{j}.

It satisfies m_{i}^{j} = \{ \begin{matrix}0 & i > j \\ #G & i = j = 1 \\ 1 & j = s \\ deg(C_{j}) & i = 1. \end{matrix}

Example 43.1.9. The matrix of incidence number for A_{4} are

1 2 3 4 5
1 12 6 4 3 1
2 0 2 0 3 1
3 0 0 1 0 1
4 0 0 0 3 1
5 0 0 0 0 1

�

Let Φ : G → S_{E} be a representation of G as permutation group and let us consider any orbit ω := {Φ(g)(x) : g ∈ G}; by restriction we thus obtain a representation G → S_{ω} which necessarily is equivalent to one of the representations defined by a C_{i}, 1 ≤ i ≤ s. Denoting, for each i , a_{i} the number of orbits ω thus equivalent to C_{i} we can therefore associate to Φ the^{7}

5 In particular H ∈ C, H′ ∈ C′ and H ⊂ H′. 6 Id est the number of elements H′ ∈ C_{j} which satisfy ρ_{j}(h)(H′) = H′ for each h ∈

H. 7 Burnside W., op. cit., p. 238

symbol [∑_{i=1}^{s} a_{i}C_{i}] denoting that the representation [Φ] is made up of a_{1} representations equivalent to [C_{1}], a_{2} representations equivalent to [C_{2}], and so on.

With modern notation we associate to Φ an element ∑_{i=1}^{s} a_{i}C_{i} in the free ℤ-module L_{G} = ∑_{i=1}^{s} ℤC_{i} with basis E.

In conclusion:

Proposition 43.1.10. The application U : Φ ↦ ∑_{i=1}^{s} a_{i}C_{i} defines a bijection between the set of all equivalence classes of the representations of G as permutation group of degree n and the set of elements ∑_{i=1}^{s} a_{i}C_{i} ∈ L_{G} such that a_{i} ∈ ℕ for each i and ∑_{i=1}^{s} a_{i} deg(C_{i}) = n.

If we moreover denote, for each j μ_{j} the mark of C_{j} in the representation Φ we have the relations

μ_{j} = ∑_{i=1}^{s} a_{i}m_{i}^{j}.

�

In the same way, if we consider the H-orbits of the elements in C_{j} and we denote a(ν)_{i}^{j} the number of such orbits consisting of ν elements^{8} such numbers are independent also on the choice of the transitive representation ρ_{j}, thus depending only on the couple (C_{i}, C_{j}).

Remark that we have the relation

∑_{ν} νa(ν)_{i}^{j} = deg(C_{j}).

Let us consider any such sequence A_{i}^{j} := (a(1)_{i}^{j}, a(2)_{i}^{j}, …, a(ν)_{i}^{j}, …) and let us remark that a(ν)_{i}^{j} ≠ 0 ⟹ ν ≤ deg(C_{j}) so that, in particular, there are at most a finite number of values ν for which a(ν)_{i}^{j} ≥ 1.

The sequences A_{i}^{j} can be stored more compactly as

B_{i}^{j} := [(a_{1}, ν_{1}), …, (a_{r}, ν_{r})] : r ≥ 1, 1 ≤ ν_{1} < ν_{2} < ⋯, a_{i} ≥ 1 for each i,

where ν_{l} are the values for which a_{l} := a(ν_{l})_{i}^{j} ≠ 0.

Definition 43.1.11. The partition array defined by G is the array, whose rows and columns are indexed by conjugacy classes of G and whose (i, j)th entry is B_{i}^{j}

Example 43.1.12. The array (A_{i}^{j}) for A_{4} is

1  2  3  4  5
1  (12, 0, …)  (6, 0, …)  (4, 0, …)  (3, 0, …)  (1, 0, …)
2  (0, 6, 0, …)  (2, 2, 0, …)  (0, 2, 0 …)  (3, 0, …)  (1, 0, …)
3  (0, 0, 4, 0, …)  (0, 0, 2, 0, …)  (1, 0, 1, 0, …)  (0, 0, 1, 0, …)  (1, 0, …)
4  (0, 0, 0, 3, 0, …)  (0, 3, 0, …)  (0, 0, 0, 1, 0, …)  (3, 0, …)  (1, 0, …)
5  (0, …, 0, 1, 0, …)  (0, …, 0, 1, 0, …)  (0, 0, 0, 1, 0, …)  (0, 0, 1, 0, …)  (1, 0, …)

8 So that, in particular m_{i}^{j} = a(1)_{i}^{j}.

and the corresponding partition array defined by A_{4} is

1  2  3  4  5
1 [(12, 1)] [(6, 1)]  [(4, 1)]  [(3, 1)] [(1, 1)]
2 [(6, 2)] [(2, 1), (2, 2)] [(2, 2)]  [(3, 1)] [(1, 1)]
3 [(4, 3)] [(2, 3)]  [(1, 1), (3, 1)] [(1, 3)] [(1, 1)]
4 [(3, 4)] [(3, 2)]  [(1, 4)]  [(3, 1)] [(1, 1)]
5 [(1, 12)] [(1, 6)]  [(1, 4)]  [(1, 3)] [(1, 1)]

Proposition 43.1.13. The rows of partition array defined by G are all different.

Proof. For any values i, j, 1 ≤ j < i ≤ s it is sufficient to show that B_{j}^{j} ≠ B_{i}^{j}. Let us fix elements H_{i} ∈ C_{i} and H_{j} ∈ C_{j} and denote N_{j} := {g ∈ G : gH_{j}g^{−1}} the normalizer of H_{j} in G. Thus the claim follows from

a(1)_{j}^{j} = [N_{j} : H_{j}] ≠ 0 = m_{i}^{j} = a(1)_{i}^{j}.

�

43.2 Representation as Permutation Group of Roots

Let f(T) := T^{n} + a_{1}T^{n−1} + ⋯ + a_{i}T^{n−i} + ⋯ + a_{n−1}T + a_{n} ∈ k[T] be a monic, separable polynomial of degree n over a field k; denote

k the algebraic closure of k, ℜ_{f} := {α_{1}, …, α_{n}} ⊂ k the set of the roots of f, K_{f} := k[α_{1}, …, α_{n}], k ⊂ K_{f} ⊂ k its splitting field, G(K_{f}/k) its Galois group

so that, in particular

f(T) = T^{n} + ∑_{i=1}^{n} a_{i}T^{n−i} = ∏_{j=1}^{n}(T − α_{j}).

We have the natural representation of G(K_{f}/k) as a permutation group of the roots of f

G(K_{f}/k) ↪ S_{ℜ_{f}} = S_{n}, σ ↦ s_{σ} : σ(α_{i}) = α_{s_{σ}(i)};

remark that

• the representation is independent on the enumeration of the roots^{9};

9 More precisely, let t ∈ S_{n} and defined β_{i} := α_{t(i)} for each i; then for each σ ∈ G(K_{f} /k) we have

σ(β_{i}) = σ(α_{t(i)}) = α_{s_{σ}t(i)} = β_{t^{−1}s_{σ}t(i)} for each i;

so a different enumeration of roots simply give an equivalent representation of G(K_{f} /k).

• G(K_{f}/k) is transitive.

Assume now that f has a factorization f = ∏_{j=1}^{r} p_{j} into irreducible components. G(K_{f}/k) then operates transitively over each ℜ_{p_{j}}; denoting, for each j, C_{i_{j}} the conjugacy class of the subgroup G(K_{p_{j}}/k) ⊂ G(K_{f}/k) we have G(K_{f}/k) = ∑_{j} C_{i_{j}} ∈ L_{G(K_{f} /k)}.

Conversely, let us assume k to be infinite and let K, k ⊂ K ⊂ k be a finite extension of k, [K : k] = n, and let us consider a faithful representation G(K/k) → S_{n} of degree n and the corresponding orbits ω_{1}, …, ω_{r} of {1, 2, …, n}; if we set n_{i} := #ω_{i}, we can reenumerate both the orbits and the elements in {1, 2, …, n} so that ω_{1} = {1, …, n_{1}}, ω_{2} = {n_{1} + 1, …, n_{2}}, …, ω_{r} = {∑_{i=1}^{r−1} n_{i} + 1, …, n}, n_{1} ≤ n_{2} ⋯ ≤ n_{r}. We can than fix any element a_{i} ∈ ω_{i}, n_{i−1} < a_{i} ≤ n_{i} and denote

G_{i} := {g ∈ G(K/k) : g(a_{i}) = a_{i}}, K_{i} := {α ∈ K : g(α) = α, g ∈ G_{i}}, ξ_{i} ∈ K_{i} a primitive element so that K_{i} = k[ξ_{i}], P_{i} ∈ k[T] its minimal polynomial, ℜ_{i} := {ξ_{i1}, …, ξ_{in_{i}}} the conjugates of ξ_{i}.

Then, since k is assumed to be infinite, we can assume that the sets ℜ_{i} are disjoint; therefore f := ∏_{i} P_{i} is separable. Clearly

{σ ∈ G(K/k) : σ(ξ_{ij}) = ξ_{ij} for each i, j} = {Id_{G(K/k)}}

so that K = k (ξ_{11}, …, ξ_{ij}, …, ξ_{rn_{r}}) .

Corollary 43.2.1. If k is infinite, for each finite extension K, k ⊂ K ⊂ k the Galois field G(K/k) can be faithfully represented as G(K_{f}/k) for a separable

polynomial f ∈ k[T].  �

43.3 Universal Lagrange Resolvent

Let us fix an integer n and let us denote

A := k[X_{1}, …, X_{n}], F := k(X_{1}, …, X_{n}), σ_{1}, ⋯, σ_{n} the elementary symmetric functions of X_{1}, …, X_{n}, I ⊂ A the ideal generated by σ_{1}, ⋯, σ_{n}, S := k[σ_{1}, ⋯, σ_{n}], K := k(σ_{1}, ⋯, σ_{n}), F (T) ∈ S[T] the polynomial

F (T) = T^{n} + ∑_{i=1}^{n}(−1)^{i}σ_{i}T^{n−i} = ∏_{j=1}^{n}(T − X_{j}).

We have

(1) F ⊃ K is an algebraic extension F = K_{F};

(2) A is the integral closure of S in F;

(3) the representation of G(F/K) ↪ S_{{X_{1},...,X_{n}}} = S_{n} as a permutation group of the roots of F is an isomorphism.

Under this isomorphism, we can therefore associate to each subgroup H ⊂ S_{n} the corresponding invariant field

I(H) := {α ∈ F : h(α) = α, for each h ∈ H}

and we have

(4) F ⊃ I(H) is an algebraic extension;

(5) I(H) is a separable extension of K;

(6) H = G(F/I(H));

(7) the integral closure of S in I(H) is A_{H} := A ∩ I(H);

(8) there are elements Ψ ∈ A_{H} which are K-primitive for I(H), id est which satisfy I(H) = K[Ψ];

(9) A_{H}, being integrally closed and noetherian, is a finite S-module;

(10) I(H) = \{\frac{a}{b} : a ∈ A_{H}, b ∈ S, b ≠ 0\} ⊂ k(σ_{1}, ⋯, σ_{n})[X_{1}, …, X_{n}] = K[X_{1}, …, X_{n}]^{10};

(11) I(H) is the fraction field of A_{H};

(12) the rank of A_{H} as a S-module is dim_{K}(I(H)) = [S_{n} : H] := \frac{n!}{#H};

(13) the finite set B := {X_{1}^{a_{1}}X_{2}^{a_{2}} ⋯ X_{n}^{a_{n}}, 0 ≤ a_{i} < i}, #B = n! is both a k-basis of A/I and a basis of A as an S-module;

On the basis of (8) we can introduce the following

Definition 43.3.1. Each K-primitive element Ψ ∈ A_{H} for I(H) is called a resolvent of H and is said to be homogeneous if it is a homogeneous polynomial in A.

The minimal polynomial L_{Ψ} ∈ S[T] of Ψ over K is called the Lagrange resolvent of H associated to Ψ.

Example 43.3.2. The Vandermonde determinant

Ψ := ∏_{i>j}(X_{j} − X_{i}) = | \begin{matrix}1 & 1 & ⋯ & 1 \\ X_{1} & X_{2} & ⋯ & X_{n} \\ X_{1}^{2} & X_{2}^{2} & ⋯ & X_{n}^{2} \\ ⋮ & ⋮ & ... & ⋮ \\ X_{1}^{n−1} & X_{2}^{n−1} & ⋯ & X_{n}^{n−1} \end{matrix} |

10 In fact if x ∈ I(H), then there are α, β ∈ A, β ≠ 0 such that x = \frac{α}{β}. It is sufficient to define

b := ∏_{σ∈S_{n}} σ(β) ∈ S \ {0}, a := α ∏_{σ∈S_{n}\{Id}} σ(β) ∈ A_{H}

in order to have x = \frac{a}{b}.

is a K-primitive element for A_{4}; its minimal polynomial is L_{Ψ} := T^{2} − Disc(F) ∈ S[T]. where Disc(F) := ∏_{i>j}(X_{j} − X_{i})^{2} is the discriminant of

the polynomial F (T) ∈ S[T].  �

Lemma 43.3.3. If Ψ ∈ A and H := {g ∈ S_{n} : g(Ψ) = Ψ} then Ψ is a

resolvent of H.  �

Remark 43.3.4. Each group H possesses homogeneous resolvents provided k is infinite. In fact, if Ψ ∈ A_{H} is a K-primitive element of degree d, denote, Ψ_{t} the homogeneization of Ψ by tσ_{1},

Ψ_{t} = (tσ_{1})^{d}Ψ(\frac{X_{1}}{tσ_{1}} , ⋯ \frac{X_{n}}{tσ_{1}} .

Consider now a set of e := [S_{n} : H] elements

{τ_{1}, …, τ_{e}} ⊂ S_{n}

such that {τ_{1}H, …, τ_{e}H} is the set of all the left classes and let us denote P_{t}(T) := ∏_{i=1}^{e}(T −τ_{i}(Ψ_{t})) and D(t) := Disc(P_{t}) ∈ S[t] its discriminant. Since P_{\frac{1}{σ_{1}}} (T) = L_{Ψ}, D(\frac{1}{σ_{1}} ) ≠ 0; thus D(t) ≠ 0 and, k being infinite, there is λ ∈ k such that D(λ) ≠ 0 so that Ψ_{λ} , which is homogeneous of degree d, is Kprimitive, thus being the required homogeneous resolvent. Its corresponding

Lagrange resolvent is L_{Ψ_{λ}} = P_{λ}(T).  �

Theorem 43.3.5 (Lagrange). Let H ⊂ S_{n} be a subgroup and Ψ ∈ A_{H} a resolvent of H. Denoting Δ_{Ψ} := Disc(L_{Ψ}) the discriminant of the Lagrange resolvent of Ψ, we have

A_{H} ⊂ \{\frac{f(Ψ)}{Δ_{Ψ}}, f ∈ S[T]\} .

Proof. Let e = [S_{n} : H] and {τ_{1}, …, τ_{e}} ⊂ S_{n} be such that {τ_{1}H, …, τ_{e}H} is the set of all the left cosets of H, with τ_{1} = Id_{S_{n}}. Set Ψ_{i} := τ_{i}(Ψ) for each i so that

L_{Ψ} = ∏_{i=1}^{e}(T − Ψ_{i}) = T^{e} + ∑_{i=1}^{e}(−1)^{i}C_{i}T^{e−i}, C_{i} ∈ S.

If we denote S_{1}, …, S_{e−1} the elementary symmetric functions on Ψ_{2}, …, Ψ_{e}, since

T^{e−1} + ∑_{j=1}^{e−1}(−1)^{j}S_{j}T^{e−j−1}

= ∏_{i=2}^{e}(T − Ψ_{i})

= \frac{L_{Ψ}}{T − Ψ_{1}}
= T^{e−1} + (Ψ_{1} − C_{1})T^{e−2} + (Ψ_{1}^{2} − C_{1}Ψ_{1} − C_{2})T^{n−i} + ⋯

is a polynomial in S[Ψ][T], we have S_{j} ∈ S[Ψ] for each j.

Consider g ∈ A_{H}, denote g_{i} := τ_{i}(g) for each i and set, for each m, 0 ≤ m < e,

h_{m} := ∑_{j=1}^{e} g_{j}Ψ_{j}^{m} = ∑_{j=1}^{e} τ_{j}(gΨ^{m}) ∈ S.

The g_{i}s can be solved à la Cremer in terms of the h_{m}: setting

D := | \begin{matrix}h_{0} & 1 & ⋯ & 1 \\ h_{1} & Ψ_{2} & ⋯ & Ψ_{e} \\ ⋮ & ⋮ & ... & ⋮ \\ h_{e−1} & Ψ_{2}^{e−1} & ⋯ & Ψ_{e}^{e−1} \end{matrix} | and δ_{Ψ} := | \begin{matrix}1 & 1 & ⋯ & 1 \\ Ψ_{1} & Ψ_{2} & ⋯ & Ψ_{e} \\ ⋮ & ⋮ & ... & ⋮ \\ Ψ_{1}^{e−1} & Ψ_{2}^{e−1} & ⋯ & Ψ_{e}^{e−1} \end{matrix} |

so that

δ_{Ψ}^{2} = ∏_{1≤i<j≤e}(Ψ_{j} − Ψ_{i})^{2} = Disc(L_{Ψ}) = Δ_{Ψ}

we have g = \frac{δ_{Ψ} D}{Δ_{Ψ}} .

We prove our claim if we show that E := δ_{Ψ} D ∈ S[Ψ]: we know that E can be expressed as E(Ψ_{2}, …, Ψ_{n}) with E ∈ S[Ψ][T_{2}, …, T_{n}] symmetric in T_{2}, …, T_{e}; thus the Fundamental Theorem on Symmetric Functions grants the existence of a polynomial F ∈ S[Ψ][Y_{1}, …, Y_{e−1}] for which

E(Ψ_{2}, …, Ψ_{n}) = F (S_{1}, …, S_{e−1}) ∈ S[Ψ]
as claimed.  �

43.4 Cauchy modules

Let us use the same notation as in Sections 43.2 and 43.3 and let us consider a monic, separable polynomial

f(T) := T^{n} + a_{1}T^{n−1} + ⋯ + a_{i}T^{n−i} + ⋯ + a_{n−1}T + a_{n} ∈ k[T]

Definition 43.4.1 (Ampère). The n interpolating functions

f_{i}(T) = f_{i}(X_{1}, …, X_{i−1}, T) ∈ k[X_{1}, …, X_{i−1}][T], 1 ≤ i ≤ n

are recursively defined as

f_{1}(T) := f(T) and f_{i}(T) := \frac{f_{i−1}(T) − f_{i−1}(X_{i−1})}{T − X_{i−1}}, 1 < i ≤ n.

�

Definition 43.4.2. The polynomials f_{i}(X_{i}) = f_{i}(X_{1}, …, X_{i}), 1 ≤ i ≤ n are

called the Cauchy modules associated to f.  �

Example 43.4.3. For n = 5 we have

f_{1}(X_{1}) = X_{1}^{5} + X_{1}^{4}a_{1} + X_{1}^{3}a_{2} + X_{1}^{2}a_{3} + X_{1}a_{4} + a_{5},
f_{2}(X_{2}) = X_{2}^{4} + X_{2}^{3}X_{1} + X_{2}^{2}X_{1}^{2} + X_{2}X_{1}^{3} + X_{1}^{4}
+ a_{1}(X_{2}^{3} + X_{2}^{2}X_{1} + X_{2}X_{1}^{2} + X_{1}^{3})
+ a_{2}(X_{2}^{2} + X_{2}X_{1} + X_{1}^{2})
+ a_{3}(X_{2} + X_{1}) + a_{4},
f_{3}(X_{3}) = X_{3}^{3} + X_{3}^{2}X_{2} + X_{3}X_{2}^{2} + X_{2}^{3} + X_{3}^{2}X_{1}
+ X_{3}X_{2}X_{1} + X_{2}^{2}X_{1} + X_{3}X_{1}^{2} + X_{2}X_{1}^{2} + X_{1}^{3}
+ a_{1}(X_{3}^{2} + X_{3}X_{2} + X_{2}^{2} + X_{3}X_{1} + X_{2}X_{1} + X_{1}^{2})
+ a_{2}(X_{3} + X_{2} + X_{1}) + a_{3},
f_{4}(X_{4}) = X_{4}^{2} + X_{4}X_{3} + X_{3}^{2} + X_{4}X_{2} + X_{3}X_{2}
+ X_{2}^{2} + X_{4}X_{1} + X_{3}X_{1} + X_{2}X_{1} + X_{1}^{2}
+ a_{1}(X_{4} + X_{3} + X_{2} + X_{1}) + a_{2},
f_{5}(X_{5}) = X_{5} + X_{4} + X_{3} + X_{2} + X_{1} + a_{1}

�

Lemma 43.4.4 (Cauchy). The Cauchy modules satisfy deg_{i}(f_{i})+i = n+1 and lc(f_{i}) = 1 for each i.

Moreover, under the further assumption that the roots are all distinct, for

each i, the roots of f_{i}(α_{1}, …, α_{i−1}, X_{i}) are {α_{j}, i ≤ j ≤ n}.  �

Proof. The claims being true by definition for i = 1 we inductively have f_{i}(α_{1}, …, α_{i−1}, α_{j}) = \frac{f_{i−1}(α_{1},...,α_{i−2},α_{j})−f_{i−1}(α_{1},...,α_{i−2},α_{i−1})}{α_{j}−α_{i−1}} = 0 for each

j, i ≤ j ≤ n.  �

Lemma 43.4.5 (Cauchy). Assume g ∈ k[X] is such that there is u ∈ k for which g(α) = u for each root α ∈ ℜ_{f} of f. Then Rem(g, f) = u.

Proof. The polynomial Rem(g, f) − u has degree less then deg(f) and van-

ishes in each root of f; therefore is zero.  �

Let g_{n}(X_{1}, …, X_{n}) ∈ A = k[X_{1}, …, X_{n−1}][X_{n}] be symmetric in the variables X_{1}, …, X_{n} and recursively denote

g_{n−1} := Rem(g_{n}, f_{n}(X_{n})) ∈ k[X_{1}, …, X_{n−1}] the remainder of the division

of g_{n} by f_{n} in k[X_{1}, …, X_{n−1}][X_{n}]; g_{n−2} := Rem(g_{n−1}, f_{n−1}(X_{n−1})) ∈ k[X_{1}, …, X_{n−2}] the remainder of the

division of g_{n−1} by f_{n−1} in k[X_{1}, …, X_{n−2}][X_{n−1}]; … g_{i} := Rem(g_{i+1}, f_{i+1}(X_{i+1})) ∈ k[X_{1}, …, X_{i}] the remainder of the division

of g_{i+1} by f_{i+1} in k[X_{1}, …, X_{i}][X_{i+1}]; … g_{1} := Rem(g_{2}, f_{2}(X_{2})) ∈ k[X_{1}] the remainder of the division of g_{2} by f_{2} in

k[X_{1}][X_{2}]; g_{0} := Rem(g_{1}, f_{1}(X_{1})) ∈ k the remainder of the division of g_{1} by f_{1} in

k[X_{1}],

remarking that we can assume g_{i} ∈ k[X_{1}, …, X_{i}] instead of the weeker g_{i} ∈ k(X_{1}, …, X_{i}) because lc(f_{i}) = 1.

Remark 43.4.6 (Cauchy). Each g_{i} is a symmetric polynomial in the variables X_{1}, …, X_{i}.

Moreover, repeatedly applying Lemma 43.4.5, if #ℜ_{f} = n, id est all the roots of f are distinct we have that, for each i, g_{i}(α_{1}, …, α_{i}) ∈ k(α_{1}, …, α_{i}) satisfies

g_{i}(α_{1}, …, α_{i}) = g_{i+1}(α_{1}, …, α_{i}, α_{j}), i < j ≤ n

so that g_{0} = g_{n}(α_{1}, …, α_{n}).

Donc alors la valeur [g_{0}] de [g_{n}(α_{1}, …, α_{n})] , determinèe comme nous l’avons dit ci-dessus, sera une fonction rationelle et même entière, par conséquent une function continue des coefficients renfermés dans f(x). D’ailleurs chacun de ces coefficients représentera, au signe près, ou la somme des racines de l’èquation [f(x) = 0], ou la somme formée avec les produits qu’on obtient en multipliant ces racines deux à deux, trois à troix, etc. Donc la valeur trouvée de [g_{0}] pourra être encore considérée comme une fonction continue des racines de l’èquation [f(x) = 0]; et dans la formule

[g_{n}(α_{1}, …, α_{n}) = g_{0}]

qui se vérifiera toutes les fois que les racines [(α_{1}, …, α_{n}] seront inégales, les deux membres varieront par degrée insensible en même temps que ces racines. [⋯] Il est mainteneant facile de s’assurer que [the result] s’etende, avec la formule [g_{n}(α_{1}, …, α_{n}) = g_{0}], au cas même oú l’èquation [f(x) = 0] offre des racines égales. Car des racines égales de l’èquation [f(x) = 0] peuvent être considérées des valeures variables de racines supposées d’abord inégales, mais trés peu différent les unes des autres; et puisque la formule [g_{n}(α_{1}, …, α_{n}) = g_{0}] , dont les deux membres varient par dégres insensibles avec les racines, par conséquent avec leurs différences, continuera de subsister pour des valeurs de ces différences aussi rapprochées de zéro que l’on voudra, elle subsisteracerainement das le cas même oú ces différences viendront à s’évanouir. ^{11}

11 A. Cauchy Usage des fonctions interpolaires dans ls determination des fonctions symmetriques des racines d’une équation algébrique donnée C.R. Acad. Sci. Paris 11 (1840) p.933,

In: A. Cauchy Oeuvres t. V, Gauthier–Villars (1882) Paris , pp. 476–7.

Thus we have:

Théorème II. Soient

f(x)

une function entière de x, du degré n, et

f(a, x) = \frac{f(x) − f(a)}{x − a}, f(a, b, x) = \frac{f(a, x) − f(a, b)}{x − b}, ⋯

les functions interpolaires de divers ordres qui renfermant avec la variable x diverses valeurs particolières a, b, c, … de cette variable. Concevons d’ailleurs que les letters

a, b, c, …, h, k

représentent les n racines de l’équarion

f(x) = 0

et désignon par

F (a, b, c, ⋯, h, k)

une function entière mais symmétrique de ces racines. Pour éliminer de cette même fonction les racines

k, h, …, c, b, a

il suffira de la diviser successivement par les divers terms de la suite

f(a, b, c, …, h, k), f(a, b, c, …, h,), …, f(a, b, c), f(a, b), f(a),

considérés le premier comme fonction de k, le second comme fonction de h,…, l’avant-dernier comme fonction de b, le dernier comme fonction de a. Le dernier des rests ainsi obtenus sera indépendent de a, b, c, …, h, k, et représentera nécessariament la valeur U de la fonction symmétrique

F (a, b, c, ⋯, h, k)

exprimée à l’aide des coefficients que renferme le premier membre de l’èquation [f(x) = 0].^{12}

In conclusion the argument above gives:

Theorem 43.4.7 (Cauchy). Let g_{n}(X_{1}, …, X_{n}) ∈ A be a symmetric polynomial in the variables X_{1}, …, X_{n} and let g_{n−1}, …, g_{0} be obtained as above by successively dividing g_{n}(X_{1}, …, X_{n}) by f_{n}(X_{n}), f_{n−1}(X_{n−1}), …, f_{1}(X_{1}).

The last remainder g_{0} will be independent of X_{1}, …, X_{n} and gives the value g_{n}(α_{1}, …, α_{n}) as a function of the coeffiecients a_{1}, …, a_{n} of f. �

12 A. Cauchy, op. cit., pp. 474–5.

Example 43.4.8. For n = 3 we have

f_{1}(X_{1}) = X_{1}^{3} + X_{1}^{2}a_{1} + X_{1}a_{2} + a_{3},
f_{2}(X_{2}) = X_{2}^{2} + X_{2}X_{1} + X_{1}^{2} + a_{1}(X_{2} + X_{1}) + a_{2},
f_{3}(X_{3}) = X_{3} + X_{2} + X_{1} + a_{1}

For the symmetric polynomial

g = X_{3}^{3}X_{2} + X_{3}X_{2}^{3} + X_{3}^{3}X_{1} + X_{2}^{3}X_{1} + X_{3}X_{1}^{3} + X_{2}X_{1}^{3}

we have

g_{2}(X_{1}, X_{2}) = −2X_{2}^{4} − 4X_{2}^{3}X_{1} − 6X_{2}^{2}X_{1}^{2} − 4X_{2}X_{1}^{3} − 2X_{1}^{4}
+ a_{1}(−4X_{2}^{3} − 9X_{2}^{2}X_{1} − 9X_{2}X_{1}^{2} − 4X_{1}^{3})
+ a_{1}^{2}(−3X_{2}^{2} − 6X_{2}X_{1} − 3X_{1}^{2}) − a_{1}^{3}(X_{2} − X_{1})
g_{1}(X_{1}) = X_{1}^{3}a_{1} + X_{1}^{2}a_{1}^{2} + X_{1}a_{1}a_{2} + a_{1}^{2}a_{2} − 2a_{2}^{2},
g_{0} = a_{1}^{2}a_{2} − 2a_{2}^{2} − a_{1}a_{3}.

�

Let us now extend the notations of Sections 43.2 and 43.3 by denoting

𝔽 the prime field of k, J = 𝕀(σ_{1} + a_{1}, σ_{2} − a_{2}, … σ_{n} − (−1)^{n}a_{n}) ⊂ A A := k[X_{1}, …, X_{n}], J^{e} := Jk[X_{1}, …, X_{n}], B := {X_{1}^{a_{1}}X_{2}^{a_{2}} ⋯ X_{n}^{a_{n}}, 0 ≤ a_{i} < i}, B′ := {X_{1}^{a_{1}}X_{2}^{a_{2}} ⋯ X_{n}^{a_{n}}, 0 ≤ a_{i} < n − i}, Γ := G(K_{f}/k) ⊂ S_{n}, R : Γ ↪ S_{ℜ_{f}} = S_{n} the canonical representation of Γ as a permutation

group of the roots of f defined by

R(u) = s_{u} : u(α_{i}) = α_{s_{u}(i)};

˜ ·: A → K_{f} ⊂ k the k-algebra morphism defined by

g̃ = g(α_{1}, …, α_{n}), for each g ∈ A.

Remark 43.4.9. For each g ∈ A and each s ∈ S_{n} we have

s̃(g) = s(g)(α_{1}, …, α_{n}) = s (g(α_{1}, …, α_{n})) = s(g̃)

where S_{n} is interepreted as

S_{n} = G(F/K) in the left hand side and S_{n} = G(K_{f}/k) in the right hand side.

Remark 43.4.10. For each g(X_{1}, …, X_{n}) ∈ A = k[X_{1}, …, X_{n}] which is symmetric in the variables X_{1}, …, X_{n} we have both • g̃ ∈ k and

• g − g̃ ∈ J.

Moreover, with the present notation Cauchy’s Theorem 43.4.7 can be read as

Let g(X_{1}, …, X_{n}) ∈ A be a symmetric polynomial in the variables X_{1}, …, X_{n}; set g_{n} := g, and let g_{n−1}, …, g_{0} be obtained by successively dividing g_{n}(X_{1}, …, X_{n}) by f_{n}(X_{n}), f_{n−1}(X_{n−1}), …, f_{1}(X_{1}). The last remainder g_{0} satisfies

g_{0} = g̃ ∈ 𝔽(a_{1}, …, a_{n}).

�

Proposition 43.4.11 (Machi–Valibouze). The reduced Gröbner basis of J (and also of J^{e}) w.r.t. any termordering < induced by X_{1} < X_{2} < … < X_{n}

is {f_{1}, …, f_{n}}.  �

Proof. Remark that for each i we have T(f_{i}) = X_{i}^{n−i+1} for any term-ordering < induced by X_{1} < X_{2} < … < X_{n}. Therefore Buchberger’s First Criterion (Lemma 22.5.1) grants that {f_{1}, …, f_{n}} is the Gröbner basis of the ideal it generates w.r.t. any such termordering.

We have therefore just to prove that

J = 𝕀(f_{1}, …, f_{n}).

Clearly, Cauchy’s Theorem grants J ⊆ 𝕀(f_{1}, …, f_{n}) and equality is a direct consequence of the remark that N(𝕀(f_{1}, …, f_{n})) = B′ so that

dim(A/J) = #B = n! = #B′ = dim (A/𝕀(f_{1}, …, f_{n})) .

�

To complete our argument, we need to consider the ”generic” monic polynomial

f(T) = T^{n} + a_{1}T^{n−1} + … + a_{n−1}T + a_{n} ∈ 𝔽(a_{1}, …, a_{n})[T]

and express the associated Cauchy modules f_{i}, which are symmetric polynomials in X_{1}, …, X_{i}, as elements in

f_{i} ∈ 𝔽(a_{1}, …, a_{n})(X_{1}, …, X_{i}).

Lemma 43.4.12. With the present notation it holds:

(1) f_{n}(T) − f_{n}(X_{n}) = T − X_{n};

(2) f(T) = f_{ν+1}(T) ∏_{j=1}^{ν}(T − X_{j}) + ∑_{i=1}^{ν} f_{i}(X_{i}) ∏_{j=1}^{i−1}(T − X_{j}) for each ν, 1 ≤ ν < n;

(3) f(T) = F (T) + ∑_{i=1}^{n} f_{i}(X_{i}) ∏_{j=1}^{i−1}(T − X_{j}); (4) f(T) − F (T) = ∑_{i=1}^{n−1} (a_{i} − (−1)^{i}σ_{i}) T^{n−i} = ∑_{i=1}^{n} f_{i}(X_{i}) ∏_{j=1}^{i−1}(T − X_{j}).

Proof. (1) requires just a trivial verification;

(2) We have

f_{i}(T)(T − X_{i−1}) = f_{i−1}(T) − f_{i−1}(X_{i−1})

so that (i = 1) f(T) = f_{1}(T) = f_{2}(T)(T − X_{1}) + f_{1}(X_{1}).

Thus, inductively,

f(T) = f_{ν}(T) ∏_{j=1}^{ν−1}(T − X_{j}) + ∑_{i=1}^{ν−1} f_{i}(X_{i}) ∏_{j=1}^{i−1}(T − X_{j})

= (f_{ν}(X_{ν}) + f_{ν+1}(T)(T − X_{ν})) ∏_{j=1}^{ν−1}(T − X_{j})

+ ∑_{i=1}^{ν−1} f_{i}(X_{i}) ∏_{j=1}^{i−1}(T − X_{j})

= f_{ν+1}(T) ∏_{j=1}^{ν}(T − X_{j}) + ∑_{i=1}^{ν} f_{i}(X_{i}) ∏_{j=1}^{i−1}(T − X_{j});

(3) (1) implies

f_{n}(T) ∏_{j=1}^{n−1}(T − X_{j}) = f_{n}(X_{n}) ∏_{j=1}^{n−1}(T − X_{j}) + ∏_{j=1}^{n}(T − X_{j})

= f_{n}(X_{n}) ∏_{j=1}^{n−1}(T − X_{j}) + F (T).

The claim than follows by substituting this result in the formula of (2) for ν := n − 1.

(4) Trivial.  �

Denoting h_{d}(X_{1}, …, X_{i}) the d^{th} complete sum in k[X_{1}, …, X_{i}], id est (Compare Definition 6.3.2) the sum of all terms of degree d in k[X_{1}, …, X_{i}] we have

Proposition 43.4.13. It holds, setting a_{0} = 1,

(1) \frac{h_{d}(X_{1},...,X_{i−2},X_{i})−h_{d}(X_{1},...,X_{i−2},X_{i−1})}{X_{i}−X_{i−1}} = h_{d−1}(X_{1}, …, X_{i}) for each d and each i ≤ n;

(2) f_{i}(X_{i}) = ∑_{d=0}^{n−i+1} a_{n−i+1−d}h_{d}(X_{1}, …, X_{i}).

Proof. (1) Trivial (2) Thus

f_{i}(X_{i})
= \frac{f_{i−1}(X_{i}) − f_{i−1}(X_{i−1})}{X_{i} − X_{i−1}}

= ∑_{d=0}^{n−i+2} a_{n−i+2−d} \frac{h_{d}(X_{1}, …, X_{i−2}, X_{i}) − h_{d}(X_{1}, …, X_{i−2}, X_{i−1})}{X_{i} − X_{i−1}}

= ∑_{d=1}^{n−i+2} a_{n−i+2−d}h_{d−1}(X_{1}, …, X_{i})

= ∑_{d=0}^{n−i+1} a_{n−i+1−d}h_{d}(X_{1}, …, X_{i}).

�

Corollary 43.4.14. (Compare Proposition 6.3.15 and Fact 6.3.14)

The Gröbner basis of I = 𝕀(σ_{1}, …, σ_{n}) w.r.t. the lex termordering < induced by X_{1} > X_{2} > … > X_{n} is

{h_{n−i+1}(X_{1}, …, X_{i}), 1 ≤ i ≤ n}.

Proof. We have just to apply Propositions 43.4.11 and 43.4.13 with a_{1} =

… = a_{n} = 0.  �

Historical Remark 43.4.15. When stating Proposition 6.3.15 and Fact 6.3.14, I was completely unaware of Propositions 43.4.13 and of Cauchy’s Theorem 43.4.7 which imply Propositions 43.4.11.

Only later I realized that they were reported in Valibouze’s Habilitation where the history is also told:

L’idéal [J] des relations symétriques est engendré par les n polynômes [σ_{1} − a_{1}, σ_{2} − a_{2}, …, σ_{n} − a_{n}]. Augustin Cauchy utilise les fonctions interpolaires introduites par Ampère pour calculer un système de générateurs qui se révéle être une base standard réduite, pour l’ordre lexicographique, de l’idéal [J]. Cette base standard sert à tester l’appartenance à l’ideal [J] et a évaluer sur k un polynôme symétrique en les racines de f. Comme elle est réduite, la base standard de [J] permet de retrouver la base naturelle de [A/J]. Cauchy calcule cette base standard pour n = 4 et en 1990 et avec Antonio Machì nout la calculons pour tout n en utilizant des séries géneratrices tronquées [⋯]. Alain Lascoux suggère une démonstration plus courte qui fait appel aux Λ-anneaux et aux différences divisées sur les S-fonctions.

13

�

13 Valibouze A., Théorie de Galois constructive, Mémoir d’Habilitation, Paris 6 (1998) p. 23.

43.5 Resolvents and Polynomial Roots

Let us consider again the monic, separable polynomial

f(T) := T^{n} + a_{1}T^{n−1} + ⋯ + a_{i}T^{n−i} + ⋯ + a_{n−1}T + a_{n} ∈ k[T]

and use freely the same notation as in Sections 43.2 and 43.3.

Definition 43.5.1. Let Ψ ∈ A_{H} be a resolvent of H ⊂ S_{n} and let

L_{Ψ} [σ_{1}, ⋯, σ_{n}, T] ∈ k[σ_{1}, ⋯, σ_{n}][T] = S[T]

be the Lagrange resolvent of H associated to Ψ.

The (H, Ψ)-Lagrange resolvent of f is the polynomial

L_{Ψ,f}[T] := L_{Ψ} [−a_{1}, ⋯, (−1)^{i}a_{i}, ⋯, (−1)^{n}a_{n}, T] ∈ k[T].

If L_{Ψ,f} is separable, one sais that Ψ is f-separable.

Proposition 43.5.2. If k is an infinite field and A ⊂ k is a ring whose fraction field is k, there is a resolvent Ψ ∈ A[X_{1}, …, X_{n}] of H for which L_{Ψ,f} is separable; moreover Ψ can be chosen homogeneous.

Proof. Since the roots α_{i} of f are distinct, the n! polynomials

(∑_{i=1}^{n} U_{i}α_{s(i)}) − 1 ∈ k[U_{1}, …, U_{n}],

where s runs among the elements of S_{n}, are all different.

Thus denoting, for each H′ ∈ (S_{n}/H)_{l},

ϕ_{H′} := ∏_{s∈H′} ((∑_{i=1}^{n} U_{i}α_{s(i)}) − 1) ∈ k[U_{1}, …, U_{n}]

we have gcd(ϕ_{H′}, ϕ_{H′′}) = 1 for each H′, H′′ ∈ (S_{n}/H)_{l}, H′ ≠ H′′.

Since A is infinite, we can choose u_{1}, …, u_{n} ∈ A such that the elements ϕ_{H′}(u_{1}, …, u_{n}), H′ ∈ (S_{n}/H)_{l} are all distinct.

For such values, we set

Ψ := ∏_{s∈H} ((∑_{i=1}^{n} u_{i}X_{s(i)}) − 1) ∈ I(H) ∩ A[X_{1}, …, X_{n}].

Its conjugates in K are the polynomials

Ψ_{H′} := ∏_{s∈H′} ((∑_{i=1}^{n} u_{i}X_{s(i)}) − 1) , H′ ∈ (S_{n}/H)_{l}

which are all distinct so that Ψ ∈ A[X_{1}, …, X_{n}] is the required resolvent of H.

We have L_{Ψ,f} = ∏_{H′∈(S_{n}/H)_{l}} (T − Ψ_{H′}(α_{1}, …, α_{n})) which is therefore separable.

The same argument as in Remark 43.3.4 proves that Ψ can be made homogeneous: denote

(Ψ_{H′})_{t} = (tσ_{1})^{deg(Ψ_{H′})}Ψ_{H′}(\frac{X_{1}}{tσ_{1}} , ⋯ \frac{X_{n}}{tσ_{1}} , P_{t}(T) := ∏_{H′∈(S_{n}/H)_{l}} (T − (Ψ_{H′})_{t}) D(t) := Disc(P_{t}) ∈ S[t] = k[σ_{1}, ⋯, σ_{n}, t], D(t) := D(a_{1}, ⋯, a_{n}, t) ∈ A[t].

We clearly have D(t) ≠ 0 so that, A being infinite, there is λ ∈ A such that D(λ) ≠ 0; thus

Θ := Ψ_{λ} ∈ I(H) ∩ A[X_{1}, …, X_{n}]

is homogeneous of degree deg(Ψ) = #(H), and its conjugates in F are

Θ_{H′} := (Ψ_{H′})_{λ} ∈ A[X_{1}, …, X_{n}], H′ ∈ (S_{n}/H)_{l}.

Denoting, for each H′ ∈ (S_{n}/H)_{l}, θ_{H′} := Θ_{H′}(α_{1}, …, α_{n}) we have L_{Θ,f} = ∏_{H′∈(S_{n}/H)_{l}} (T − θ_{H′}) whose discrimiant satisfies D(λ) ≠ 0.

Therefore

• the Θ_{H′}s are all distinct,

• Θ is a homogeneous resolvent of H,

• L_{Θ,f} is separable.  �

Let us now denote

Z := Z(J^{e}) = {(β_{1}, …, β_{n}) ∈ k^{n} : p(β_{1}, …, β_{n}) = 0, for each p ∈ J^{e}} ⊂ k^{n}; Γ := G(K_{f}/k) ⊂ S_{n}, H := (S_{n}/Γ)_{r} := {Γ s : s ∈ S_{n}} the set of the right classes of Γ :=

G(K_{f}/k) ⊂ S_{n}, N := #Γ.

Lemma 43.5.3. It holds

(1) Z = {(α_{s(1)}, …, α_{s(n)}) : s ∈ S_{n}};

(2) A/J ≅ Span_{k}(B′), A/J^{e} ≅ Span_{k}(B′);

(3) both J and J^{e} are radical.

Proof. (1) is obvious; (2) is a trivial consequence of Proposition 43.4.11; (3) follows from

dim_{k}(A/\sqrt{J}) = #Z = n! = dim_{k}(A/J).

�

For each Γ′ ∈ H let us denote

W_{Γ′} := {(α_{s(1)}, …, α_{s(n)}) : s ∈ Γ′}, P_{Γ′} := ∏_{s∈Γ′} (T − ∑_{i=1}^{n} U_{i}α_{s(i)}); 𝔪_{Γ′} := I(W_{Γ′}) = {g ∈ A : g(β_{1}, …, β_{n}) = 0 for each (β_{1}, …, β_{n}) ∈ W_{Γ′}}.

We also denote W := W_{Γ}, Ψ := P_{Γ}, 𝔪 := 𝔪_{Γ}.

Lemma 43.5.4. With the present notation, it holds

(1) the u-resultant of Z

Ψ_{Z} := ∏_{s∈S_{n}} (T − ∑_{i=1}^{n} U_{i}α_{s(i)}) ∈ k[U_{1}, …, U_{n}][T]

factorizes into irreducible complonents as Ψ_{Z} = ∏_{Γ′∈H} P_{Γ′};

(2) the irreducible components of Z are the W_{Γ′}s : Z = ⋃_{Γ′∈H} W_{Γ′};

(3) each 𝔪_{Γ′} is maximal in A;

(4) J = ⋂_{Γ′∈H} 𝔪_{Γ′};

(5) 𝔪_{Γ′} = Z(W_{Γ′}) for each Γ′ ∈ H;

(6) K_{f} ≅ A/𝔪_{Γ′} for each Γ′ ∈ H.  �

Lemma 43.5.5 (Arnaudiès–Valibouze). Let g ∈ 𝔪, g ∉ ⋂_{{Γ′∈H \atop Γ′≠Γ}} 𝔪_{Γ′}.

Then 𝔪 is generated by

{σ_{1} + a_{1}, σ_{2} − a_{2}, …, σ_{n} − (−1)^{n}a_{n}, g}.

Proof. Denoting 𝔞 := J + (g), by assumption we have Z(𝔞) = W = Z(𝔪), whence

(1) there is ρ ∈ ℕ for which^{14} 𝔪^{ρ} ⊂ 𝔞,

(2) J ⊂ 𝔞 ⊂ \sqrt{𝔞} = 𝔪.

Since ( ⋂_{{Γ′∈H \atop Γ′≠Γ}} 𝔪_{Γ′} ) + 𝔪^{ρ} = A there are u ∈ ( ⋂_{{Γ′∈H \atop Γ′≠Γ}} 𝔪_{Γ′} ) and v ∈ 𝔪^{ρ} for

which 1 = u + v. Therefore for each x ∈ 𝔪, we have x = xu + xv with

xu ∈ 𝔪^{ρ} ⊂ 𝔞, xv ∈ 𝔪 ( ⋂_{{Γ′∈H \atop Γ′≠Γ}} 𝔪_{Γ′} ) = ⋂_{Γ′∈H} 𝔪_{Γ′} = J ⊂ 𝔞

so that x ∈ 𝔞.  �

Let Θ be a resolvent of a subgroup H of S_{n} and denote

• θ := Θ̃ = Θ(α_{1}, …, α_{n});

• Θ_{1} = Θ, Θ_{2}, …, Θ_{ν} the distinct conjugates s(Θ), s ∈ S_{n}, of Θ in F ⊃ K;

• H_{i} := {s ∈ S_{n} : s(Θ) = Θ_{i}}, 1 ≤ i ≤ ν.

14 A is noetherian.

Remarking that θ is a root of the (H, Θ)-Lagrange resolvent L_{Θ,f}[T], let us assume that

(1) θ is a simple root of L_{Θ,f}[T], and wlog

(2) its conjugates in K_{f} are θ, Θ̃_{2}, …, Θ̃_{r}, 1 ≤ r ≤ ν,

and denote

• h := ∏_{i=1}^{r}(T − Θ̃_{i}) the monic irreducible factor of L_{Θ,f}[T] in k[T];

• O := {Θ_{1}, …, Θ_{r}};

• S := {s ∈ S_{n} : s(Θ_{i}) ∈ O, for each i, 1 ≤ i ≤ r};

• g := h(Θ) ∈∈ A = k[X_{1}, …, X_{n}] for each i, 1 ≤ i ≤ r.

Theorem 43.5.6 (Arnaudiès–Valibouze). With the present notation we have

(1) The Γ -orbit of Θ in F is O,

(2) Γ ⊂ S ⊂ ⋃_{i=1}^{r} H_{i} and [S : Γ] = [S ∩ H : Γ ∩ H],

(3) Γ = S = ⋃_{i=1}^{r} H_{i} ⟺ 𝔪 = 𝕀 (σ_{1} + a_{1}, σ_{2} − a_{2}, … σ_{n} − (−1)^{n}a_{n}, g) .

Proof.

(1) For s ∈ Γ, since h ∈ k[T], we have

h({ \atop s(Θ)}) = h(s(Θ̃)) = s(h(Θ̃)) = s(0) = 0

therefore s(Θ) ∈ O.

For each i, 1 ≤ i ≤ r, since h is irreducible, there is s ∈ Γ for which { \atop s(Θ)} = s(Θ̃) = Θ̃_{i}; since, with the same argument above, we have h({ \atop s(Θ)}) = 0 then necessarily s(Θ) = Θ_{i}.

(2) The inclusion S ⊂ ⋃_{i=1}^{r} H_{i} being trivial, let us prove Γ ⊂ S: for s ∈ Γ

we have

s ({Θ̃_{1}, …, Θ̃_{r}}) = {Θ̃_{1}, …, Θ̃_{r}};

therefore the same argument as above allows to deduce

s ∈ Γ ⟹ s(Θ̃_{i}) = { \atop s(Θ_{i})} for each i, 1 ≤ i ≤ r,
⟹ s(Θ̃_{i}) ∈ {Θ̃_{1}, …, Θ̃_{r}} for each i, 1 ≤ i ≤ r,
⟹ s(Θ_{i}) ∈ {Θ_{1}, …, Θ_{r}} for each i, 1 ≤ i ≤ r,
⟹ s ∈ S.

Moreover

• O is the S-orbit of Θ,

• {s ∈ S : s(Θ) = Θ} = S ∩ H,

• {s ∈ Γ : s(Θ) = Θ} = Γ ∩ H, whence r = [S : S ∩ H] = [Γ : Γ ∩ H] so that

[S : S ∩ H][S ∩ H : Γ ∩ H] = [S : Γ][Γ : Γ ∩ H]

gives the required [S : Γ] = [S ∩ H : Γ ∩ H].

(3) Clearly g(β_{1}, …, β_{n}) = 0 for each (β_{1}, …, β_{n}) ∈ W_{Γ}. Therefore, Lemma 43.5.5 reduces the statement to Γ = S = ⋃_{i=1}^{r} H_{i} iff

g(β_{1}, …, β_{n}) ≠ 0 for each (β_{1}, …, β_{n}) ∈ ⋃_{{Γ′∈H \atop Γ′≠Γ}} W_{Γ′}.

We have

g(β_{1}, …, β_{n}) ≠ 0 for each (β_{1}, …, β_{n}) ∈ ⋃_{{Γ′∈H \atop Γ′≠Γ}} W_{Γ′}

⟺ h(Θ(α_{s(1)}, …, α_{s(n)})) ≠ 0 for each s ∈ S_{n} \ Γ
⟺ h({ \atop s(Θ)}) ≠ 0 for each s ∈ S_{n} \ Γ.

Since h is a simple factor of L_{Θ,f}[T] the only s ∈ S_{n} for which h({ \atop s(Θ)}) = 0 are those satisfying s(Θ) ∈ O id est the elements in ⋃_{i=1}^{r} H_{i}. Thus

𝔪 = 𝕀 (σ_{1} + a_{1}, σ_{2} − a_{2}, … σ_{n} − (−1)^{n}a_{n}, g)
⟺ g(β_{1}, …, β_{n}) ≠ 0 for each (β_{1}, …, β_{n}) ∈ ⋃_{{Γ′∈H \atop Γ′≠Γ}} W_{Γ′}

⟺ h({ \atop s(Θ)}) ≠ 0 for each s ∈ S_{n} \ Γ
⟺ s ∉ ⋃_{i=1}^{r} H_{i} implies s ∈ S_{n} \ Γ

⟺ ⋃_{i=1}^{r} H_{i} ⊂ Γ

which, by (2) is equivalent to Γ = S = ⋃_{i=1}^{r} H_{i}.

43.6 Lagrange resolvent and Galois group

Let

• H be a subgroup of S_{n},

• e := [S_{n} : H],

• Θ a resolvent of H,

• Θ = Θ_{1}, …, Θ_{e} its distinct conjugates in F over K,

• H_{i} := {s ∈ S_{n} : s(Θ_{1}) = Θ_{i}}, 1 ≤ i ≤ e, • θ := Θ̃,

• ν the multiplicity of θ in L_{Θ,f}[T],

so that

(A) L_{Θ}[T] = ∏_{i=1}^{e} (T − Θ_{i}) , (B) L_{Θ,f}[T] = ∏_{i=1}^{e} (T − Θ̃_{i}) , (C) (S_{n}/H)_{l} = {H_{i}, 1 ≤ i ≤ e}, H_{1} = H;

we wlog assume that

(D) Θ̃_{i} = θ ⟺ i ≤ ν,

and denote

• O := {Θ_{1}, …, Θ_{ν}};

• S := {s ∈ S_{n} : s(Θ_{i}) ∈ O, 1 ≤ i ≤ ν}.

Proposition 43.6.1. With the present notation

(1) if θ is a simple root of L_{Θ,f}[T] then

G(K_{f}/k(θ)) = {s ∈ G(K_{f}/k) : s(θ) = θ} = G(K_{f}/k) ∩ H;

(2) if ν > 1, then

G(K_{f}/k(θ) = {s ∈ G(K_{f}/k) : s(θ) = θ} = G(K_{f}/k) ∩ S;

moreover

(a) [k(θ) : k] = [G(K_{f}/k) : G(K_{f}/k) ∩ S],

(b) G(K_{f}/k) ∩ S ⊃ G(K_{f}/k) ∩ H and

(c) [G(K_{f}/k) : G(K_{f}/k) ∩ H] ≤ ν[k(θ) : k].

Proof.

(1) For each s ∈ G(K_{f}/k) and each p ∈ A we have (Remark 43.4.9) s̃(p) = s(p̃); so for each s ∈ G(K_{f}/k) ∩ H we have

θ = Θ̃ = { \atop s(Θ)} = s(Θ̃) = s(θ)

so that G(K_{f}/k) ∩ H ⊂ G(K_{f}/k(θ)).

If, instead s ∈ G(K_{f}/k) \ H, Θ′ := s(Θ) is a conjugate of Θ in F over K distinct from θ; since S_{n} = G(F/K) and L_{Θ,f}[T] = ∏_{s∈S_{n}} (T − { \atop s(Θ)}) necessarily

s(θ) = s(Θ̃) = { \atop s(Θ)} = Θ̃′ ≠ Θ̃ = θ

and s ∉ G(K_{f}/k(θ)).

(2) For each s ∈ G(K_{f}/k) we have s ∈ {s ∈ G(K_{f}/k) : s(θ) = θ} iff, for each

i, 1 ≤ i ≤ ν, { \atop s(Θ_{i})} = θ which is equivalent to { \atop s(Θ_{i})} ∈ O; thus s ∈ S and the claim. Moreover:

(a) since [K_{f} : k(θ)] = #G(K_{f}/k(θ)) = #(G(K_{f}/k) ∩ S) we have

[k(θ) : k] = \frac{[K_{f} : k]}{[K_{f} : k(θ)]} = \frac{#G(K_{f} /k)}{#(G(K_{f} /k) ∩ S)} = [G(K_{f} /k) : G(K_{f} /k) ∩ S];

(b) s ∈ H ⟹ s(Θ_{1}) = Θ_{1} ⟹ s ∈ S so that

G(K_{f}/k) ∩ H ⊂ G(K_{f}/k) ∩ S;

(c) also we have

G(K_{f}/k) ∩ S = {s ∈ G(K_{f}/k) : s(Θ_{i}) ∈ O, 1 ≤ i}

and G(K_{f}/k) ∩ H = {s ∈ S : s(Θ) = Θ}; thus

[G(K_{f}/k) ∩ S : G(K_{f}/k) ∩ H] = #{s(Θ) : s ∈ G(K_{f}/k) ∩ S}
≤ #{s(Θ) : s ∈ G(K_{f}/k) ∩ H}
= ν

and

[G(K_{f}/k) : G(K_{f}/k) ∩ H]
= [G(K_{f}/k) : G(K_{f}/k) ∩ S][G(K_{f}/k) ∩ S : G(K_{f}/k) ∩ H]
≤ [k(θ) : k]ν.

�

Corollary 43.6.2. With the present notation

(1) if ν = 1 then the degree of θ over k is [G(K_{f}/k) : G(K_{f}/k) ∩ H];

(2) θ ∈ k ⟺ G(K_{f}/k) ⊆ S;

(3) H_{i} ∩ G(K_{f}/k) ⊂ S for each i ≤ ν;

(4) H_{i} ∩ G(K_{f}/k) = ∅ for each i > ν;

(5) G(K_{f}/k) ∩ S = ⋃_{i=1}^{ν} H_{i} ∩ G(K_{f}/k).  �

Let us denote

• h(T) ∈ k[T] the monic irreducible component of L_{Θ,f} for which h(θ) = 0,

• θ_{1} = θ, θ_{2}, …, θ_{d} the G(K_{f}/k)-conugates of θ;

• R := {Θ_{1}, …, Θ_{e}},

• S(Φ) := {s ∈ S_{n} : s(Φ) = Φ} for each Φ ∈ R,

• R_{i} := {Φ ∈ R : Φ̃ = θ_{i}}, 1 ≤ i ≤ d,

• S_{i} := {s ∈ S_{n} : s(Φ) ∈ R_{i}, for each Φ ∈ R_{i}},

• O the set of the G(K_{f}/k)-orbits of R = ⋃_{i=1}^{d} R_{i};

so that

(E) h^{ν} | L_{Θ,f}, h^{ν+1} ∤ L_{Θ,f}, (F) h(T) = ∏_{i=1}^{d} (T − θ_{i}), (G) S_{1} = S, R_{1} = O, (H) #R_{i} = ν, and (Proposition 43.6.1) d = [G(K_{f}/k) : G(K_{f}/k) ∩ S_{i}] for

each i,

For each Ω ∈ O, {Ω ∩ R_{i}, 1 ≤ i ≤ d} is the set, for each j, of the G(K_{f}/k) ∩ S_{j} orbits of Ω; thus such set is independent on the choice of j but depends only on Ω. Thus we have, for each i, 1 ≤ i ≤ d, each j, 1 ≤ j ≤ d, and each Φ ∈ Ω ∩ R_{j}

m_{Ω} := #(Ω ∩ R_{i}) = [G(K_{f}/k) ∩ S_{j} : G(K_{f}/k) ∩ S(Φ)];

as a consequence

(I) #Ω = dm_{Ω} for each Ω ∈ O, (J) ∑_{Ω∈O} m_{Ω} = #R_{1} = ν,

(K) h^{ν}(T) = ∏_{Ω∈O} ∏_{Φ∈Ω} (T − Φ̃) .

Let us now introduce a second resolvent Ψ of the same subgroup H and set Ψ_{i} := s(Ψ), s ∈ H_{i}.

Definition 43.6.3. Two monic (not necessarily irreducible nor squarefree) factors F and G of, respectively, L_{Θ,f} and L_{Ψ,f} are said parallel iff there is J ⊂ {1, …, e} for which

F (T) = ∑_{j∈J} (T − Θ̃_{j}) and G(T) = ∑_{j∈J} (T − Ψ̃_{j}) .

�

If F and G are parallel and either L_{Θ,f} or L_{Ψ,f} is f-separable then J is unique.

Denoting, for each Ω ∈ O

• J_{Ω} := {j : 1 ≤ j ≤ e, Θ_{j} ∈ Ω, },

• P_{Ω}(T) := ∏_{j∈J_{Ω}} (T − Ψ̃_{j}) ,

by definition, the factor in L_{Ψ,f} parallel to h^{ν}(T) is

∏_{Ω∈O} ∏_{Ψ∈Ω} (T − Ψ̃) = ∏_{Ω∈O} ∏_{j∈J_{Ω}} (T − Ψ̃_{j}) = ∏_{Ω∈O} P_{Ω}(T).

Moreover for each Ω ∈ O, {Ψ_{j} : j ∈ J_{O}} is a G(K_{f}/k)-orbit, so that

(L) P_{Ω}(T) ∈ k[T] for each Ω ∈ O, (M) deg(P_{Ω}) = #Ω = dm_{Ω}.

In conclusion of this tour de force, we have Theorem 43.6.4 (Arnaudiès–Valibouze). Under the present notation, the factor of L_{Ψ,f} which is parallel to h^{ν} is ∏_{Ω∈O} P_{Ω}; moreover for each Ω ∈ O the polynomial P_{Ω}(T) is an irreducible factor in k[T] of L_{Ψ,f}; more-

over deg(h) | deg(P_{Ω}).  �

Let us impose on each ℕ^{e}, e ∈ ℕ \ {0}, the ordering ⪯ defined by

(b_{1}, …, b_{e}) ⪯ (a_{1}, …, a_{e}) ⟺ b_{i} ≤ a_{i} for each i.

Let us fix a subgroup H of S_{n} and let us denote (using the same notation as Section 43.1)

• E the set of all conjugacy classes of the subgroups of S_{n},

• C_{i} ∈ E the conjugacy class to which G(K_{f}/k) belongs,

• C_{j} ∈ E the conjugacy class to which H belongs,

• e := [S_{n} : H],

• (a_{1}, …, a_{e}) the sequence such that A_{i}^{j} := (a_{1}, …, a_{e}, 0, …, 0, …).

Theorem 43.6.5. With such notation, let Θ be a resolvent of a subgroup H of S_{n}; let us assume that each irreducible factor of L_{Θ,f} is separable and let us denote, for each j, 1 ≤ j ≤ e, b_{j} the number of irreducible factors of L_{Θ,f}. Then

(1) (b_{1}, …, b_{e}) ⪯ (a_{1}, …, a_{e}),

(2) if L_{Θ,f} is separable, id est Θ is f-separable, then

(b_{1}, …, b_{e}) = (a_{1}, …, a_{e}).

Proof. Let {τ_{1}, …, τ_{e}} ⊂ S_{n} be such that {τ_{1}H, …, τ_{e}H} is the set of all the left cosets of H, with τ_{1} = Id_{S_{n}}. Denote, for each i, H_{i} := τ_{i}Hτ_{i}^{−1}, Θ_{i} := τ_{i}(Θ) and θ_{i} := Θ̃_{i} so that H_{1} = H and Θ_{1} = Θ. We have

L_{Θ} = ∏_{i=1}^{e} (T − Θ_{i}) ,

L_{Θ,f} = ∏_{i=1}^{e} (T − θ_{i}) ,
H_{i} = {τ ∈ S_{n} : τ(Θ_{i}) = Θ_{i}}.

Let us fix a value j, 1 ≤ j ≤ e, and a k-irreducible simple factor p of L_{Θ,f}, deg(p) = j; then p = ∏_{i∈J} (T − θ_{i}) for some J ⊂ {1, …, e}, #J = j.

If i ∈ J, by Corollary 43.6.2.(1), j = deg(p) = [G(K_{f}/k) : G(K_{f}/k) ∩ H_{i}]; let us therefore denote, for each j,

n_{j} := #{i : 1 ≤ i ≤ e, j = [G(K_{f}/k) : G(K_{f}/k) ∩ H_{i}]}.

Clearly we have both jb_{j} ≤ n_{j} and A_{i}^{j} := (n_{1}, \frac{n_{2}}{2} , …, \frac{n_{e}}{e} , 0, …, 0, …) which proves (1).

If, moreover L_{Θ,f} is separable, we have

∑_{j=1}^{e} jb_{j} = deg(L_{Θ,f}) = e = ∑_{j=1}^{e} j \frac{n_{j}}{j}

and, since jb_{j} ≤ n_{j} = j \frac{n_{j}}{j} for each j, we obtain (2).  �

43.7 Computing Galois groups of a polynomial

Given a value n let us denote

• E := {C_{1}, …, C_{s}}, the set of all the conjugacy classes of S_{n},

• for each j, 1 ≤ j ≤ s, – H_{j} a subgroup H_{j} ∈ C_{j},

– e_{j} := [S_{n} : H_{j}]

– τ_{j1} = Id_{S_{n}}, …, τ_{je_{j}} ∈ S_{n} elements such that {τ_{j1}H_{j}, …, τ_{je_{j}}H_{j}} is the set of all the left cosets of H_{j},

– Θ_{j} a resolvent of H_{j},

– Θ_{jm} := τ_{jm}(Θ_{j}), 1 ≤ m ≤ e_{j},

– L_{Θ_{j}} := ∏_{m=1}^{e_{j}} (T − Θ_{jm}) .

If L_{Θ_{j},f} is f-separable, we denote, for each j ≤ s

• a_{jm} the number of irreducible factors of L_{Θ_{j},f} whose degree is m, for each m, 1 ≤ m ≤ e_{j};

• π(Θ_{j}, f) = (a_{j1}, …, a_{je_{j}}).

Then combining Proposition 43.1.13 and Theorem 43.6.4 we obtain

Theorem 43.7.1. With the present notation and under the assumption that all resolvents Θ_{j}, 1 ≤ j ≤ s are f-separable, then the conjugacy class of G(K_{f}/k) is C_{r} where r denotes the index of the row of the partition array B_{i}^{j} coinciding with the array

(π(Θ_{1}, f), π(Θ_{2}, f), …, π(Θ_{s}, f)) .

�

This gives an effective method for computing the conjugacy class of G(K_{f}/k); it requires to

(1) determine the partition array B_{i}^{j},

(2) choose suitable f-separable resultants Θ_{i},

(3) compute L_{Θ_{j},f},

(4) factorize them deducing the values π(Θ_{j}, f), and

(5) deduce the conjugacy class of G(K_{f}/k) by comparing the values of the partition array B_{i}^{j}.

Let us remark that

• steps (4) and (5) don’t require any special comment;

• we briefly discuss step (3) in Algorithm 43.7.2;

• the hard task is steps (1) and (2) which have been systematically solved by Arnaudiès and Valibouze for n ≤ 11^{15}.

Algorithm 43.7.2. The computation of L_{Θ_{j},f} is a direct application of the results of Proposition 43.4.11; it is sufficient to compute the Gröbner basis G of the ideal generated by

{f_{1}, ⋯, f_{n}, T − Θ_{j}} ⊂ k[T, X_{1}, …, X_{n}]

w.r.t. the lex ordering induced by T < X_{1} < X_{2} … < X_{n}; then we have

{L_{Θ_{j},f}} = G ∩ k[T].

�

We report here the results by Arnaudiès—Valibouze for n = 5; the following table lists the 11 conjugacy classes of S_{4}, reporting for a chosen element H_{j} ∈ C_{j}, their structure, their generators and their order:

H_{1} I_{4}  []  1
H_{2} S_{2}  [(3, 4)]  2
H_{3} S_{2}  [(1, 2)(3, 4)]  2
H_{4} A_{3}  [(1, 2, 3)]  3
H_{5} S_{2} × S_{2} [(1, 2), (3, 4)]  4
H_{6} V_{4}  [(1, 2)(3, 4), (1, 3)(2, 4)]  4
H_{7} ℤ_{4}  [(1, 2)(3, 4), (1, 3, 2, 4)]  4
H_{8} S_{3}  [(2, 3, 4), (3, 4)]  6
H_{9} D_{4}  [(3, 4), (1, 2)(3, 4), (1, 3)(2, 4)] 8
H_{10} A_{4}  [(1, 2)(3, 4), (1, 3)(2, 4), (2, 3, 4)] 12
H_{11} S_{4}  [(1, 4), (2, 4), (3, 4)]  24

where A_{n} denotes the alternative group, D_{n} the dihedral group, V_{4} the Viergruppe, I_{n} := {Id_{S_{n}}}. The corresponding resultants are^{16}

Θ_{1} = u_{1}X_{1} + u_{2}X_{2} + u_{3}X_{3},
Θ_{2} = X_{1} + X_{3}X_{4},
Θ_{3} = X_{1}X_{2} + X_{1}X_{3} + X_{2}X_{4},
Θ_{4} = (X_{2} − X_{3})(X_{3} − X_{4})(X_{4} − X_{2}),

15 Here I report their result for n = 4; for 5 ≤ n ≤ 11 I refer to the survey Valibouze A., Computation of the Galois Groups of the Resolvent Factors for the Deirect and Inverse Galois problems L. N. Comp. Sci. 948 (1995), 456–468, Springer, and to the LITP reports quoted there.

16 where u_{1}, u_{2}, u_{3} are distinct and non zero values.

43.7 Computing Galois groups of a polynomial 223

Θ_{5} = X_{1}X_{2},
Θ_{6} = X_{1}X_{3} + X_{2}X_{4} − X_{1}X_{2} − X_{3}X_{4},
Θ_{7} = X_{1}X_{2}^{2} + X_{2}X_{3}^{2} + X_{3}X_{4}^{2} + X_{4}X_{1}^{2},
Θ_{8} = X_{1},
Θ_{9} = X_{1}X_{2} + X_{3}X_{4},
Θ_{10} = ∏_{1≤i<j≤4}(X_{j} − X_{i}),
Θ_{11} = 1.

Finally the partition array is report in Figure 43.1.

Fig. 43.1. Partition Array for S_{4}

1  2  3  4  5  6
1  [(24, 1)]  [(12, 1)]  [(12, 1)]  [(8, 1)]  [(6, 1)]  [(6, 1)]
2  [(12, 2)]  [(2, 1), (5, 2)]  [(6, 2)]  [(4, 2)]  [(2, 1), (2, 2)]  [(3, 2)]
3  [(12, 2)]  [(6, 2)]  [(4, 1), (4, 2)]  [(4, 2)]  [(2, 1), (2, 2)]  [(6, 1)]
4  [(8, 3)]  [(4, 3)]  [(4, 3)]  [(2, 1), (2, 3)]  [(2, 3)]  [(2, 3)]
5  [(6, 4)]  [(2, 2), (2, 4)]  [(2, 2), (2, 4)]  [(2, 4)]  [(2, 1), (1, 4)]  [(3, 2)]
6  [(6, 4)]  [(3, 4)]  [(6, 2)]  [(2, 4)]  [(3, 2)]  [(6, 1)]
7  [(6, 4)]  [(3, 4)]  [(2, 2), (2, 4)]  [(2, 4)]  [(1, 2), (1, 4)]  [(3, 2)]
8  [(4, 6)]  [(2, 3), (1, 6)]  [(2, 6)]  [(1, 2), (1, 6)]  [(2, 3)]  [(1, 6)]
9  [(3, 8)]  [(1, 4), (1, 8)]  [(3, 4)]  [(8, 1)]  [(1, 2), (1, 4)]  [(3, 2)]
10  [(2, 12)]  [(1, 12)]  [(2, 6)]  [(2, 4)]  [(1, 6)]  [(2, 3)]
11  [(1, 24)]  [(1, 12)]  [(1, 12)]  [(1, 8)]  [(1, 6)]  [(1, 6)]
7  8  9  10  11
1  [(6, 1)]  [(4, 1)]  [(3, 1)]  [(2, 1)]  [(1, 1)]
2  [(3, 2)]  [(2, 1), (1, 2)]  [(1, 1), (1, 2)]  [(1, 2)]  [(1, 1)]
3  [(2, 1), (2, 2)]  [(2, 2)]  [(3, 1)]  [(2, 1)]  [(1, 1)]
4  [(2, 3)]  [(1, 1), (1, 3)]  [(1, 3)]  [[(2, 1)]  [(1, 1)]
5  [(1, 2), (1, 4)]  [(2, 2)]  [(1, 1), (1, 2)]  [(1, 2)]  [(1, 1)]
6  [(3, 2)]  [(1, 4)]  [(3, 1)]  [(2, 1)]  [(1, 1)]
7  [(2, 1), (1, 4)]  [(1, 4)]  [(1, 1), (1, 2)]  [(1, 2)]  [(1, 1)]
8  [(1, 6)]  [(1, 1), (1, 3)]  [(1, 3)]  [(1, 2)]  [(1, 1)]
9  [(1, 2), (1, 4)]  [(1, 4)]  [(1, 1), (1, 2)]  [(1, 2)]  [(1, 1)]
10  [(1, 6)]  [(1, 4)]  [(1, 3)]  [(2, 1)]  [(1, 1)]
11  [(1, 6)]  [(1, 4)]  [(1, 3)]  [(1, 2)]  [(1, 1)]

This table is applied as follows: one

• computes the discriminant L_{Θ_{10},f} of f and checks whether L_{Θ_{10},f} is a square in which case

G(K_{f}/k) = H_{j}, j ∈ {1, 3, 4, 6, 10};

• computes a factorization of L_{Θ_{8},f}; if π(Θ_{8}, f) is – [(4, 1)] ⟹ G(K_{f}/k) = H_{1},

– [(2, 1), (1, 2)] ⟹ G(K_{f}/k) = H_{2},

– [(2, 2)] and L_{Θ_{10},f} is a square ⟹ G(K_{f}/k) = H_{3},

– [(2, 2)] and L_{Θ_{10},f} is not a square ⟹ G(K_{f}/k) = H_{5},

– [(1, 1), (1, 3)] and L_{Θ_{10},f} is a square ⟹ G(K_{f}/k) = H_{4},

– [(1, 1), (1, 3)] and L_{Θ_{10},f} is not a square ⟹ G(K_{f}/k) = H_{8},

– [(1, 4)], then computes a factorization of L_{Θ_{9},f}; if π(Θ_{9}, f) is ∘ [(1, 3)] and L_{Θ_{10},f} is a square ⟹ G(K_{f}/k) = H_{10}, ∘ [(1, 3)] and L_{Θ_{10},f} is not a square ⟹ G(K_{f}/k) = H_{11}, ∘ [(3, 1)] ⟹ G(K_{f}/k) = H_{6}, ∘ [(1, 1), (1, 2)] then computes a factorization of L_{Θ_{4},f}^{17}; if π(Θ_{4}, f) is

⋄ [(1, 8)] ⟹ H_{9}, ⋄ [(2, 4)] ⟹ H_{7}.

17 one could similarly use L_{Θ_{i},f} , i ∈ {1, 2, 3, 7}.

44. Kronecker IV

The main effort in the research toward solving technques has always been devoted to ’practical’ complexity, namely smooth and fast software tools^{1} while ’theoretical’ complexity has never been deeply considered. The noteworthing exception is the TERA group based in École Polytechnique, Buenos Aires and Santander around Marc Giusti, Joos Heintz and Luis M.Pardo which in a series of papers produced in the Nineties^{2} devised a solver with good complexity. The input is assumed to be a finite set of polynomials generating a zero-dimensional ideal J ⊂ Q and given by a straight-line program, the output being

• a system of coordinates in Noetherian position for the ideal,

• a primitive element of Q/J,

• its minimal polynomial q(T) = g_{0}(T) ∈ K[T] and

• either – an Allgemaine Basis (g_{0}(T), Z_{1} − g_{1}(T), …, Z_{r} − g_{r}(T)) or

– a Kronecker/RUR presentation

q(T), \frac{∂q}{∂T} (T)Z_{1} − w_{1}(Y_{1}, …, Y_{d}, T), ⋯, \frac{∂q}{∂T} (T)Z_{r} − w_{r}(T).

1 The most effort within the PoSSo group was devoted toward an efficient memory

management! 2 Of which here and in the Bibliography I quoted only the most relevant ones:

• Giusti M., Heintz J., Morais J.E., Pardo L.M., When Polynomial Equation Systems can be “Solved” Fast?, L. N. Comp. Sci. 948 (1995), 205–231, Springer

• Giusti M., Heintz J., Morais J.E., Morgensten J., Pardo L.M., Straight-line programs in geometric elimination theory, J. Pure Appl. Algebra 124 (1998), 101– 146

• Giusti M., Heintz J., Hägele K., Morais J.E., Pardo L.M., Montaña Lower bounds for diophantine approximation, J. Pure Appl. Algebra 117–118 (1997), 277–311

• Giusti M., Heintz J., Morais J.E., Pardo L.M., Le rôle des structures de données dans les problèmes d’élimination, C.R. Acad. Sci. Paris 325 (1997), 1223–1228

• Morais J.E., Resolución eficaz de systemas de ecuaciones polinomiales, Ph. D. Thesis, Univ. Cantabria, Santander (1997)

• Giusti M., Lecerf G., Salvy B., A Gröbner Free Alternative for Polynomial System Solving, J. of Complexity 17 (2001), 154–211

• Lecerf G., Une alternative aux méthodes de réécriture pour résolution des systémes algébriques Ph.D. Thesis, École Polytechnique (2001)

The relevant result is that such algorithm has (low) polynomial complexity wrt the natural misure of the data (number of variables, degree of imput polynomials, number of roots, size of the straight-line program).

What is amazing is that this good-complexity theoretical result has produced a software solver whose practical performances compare with the best available Gröbner-based solvers.

After posing the problem approached by the TERA group (Section 44.1), discussing the technical tools, mainly an appropriate Newton-Hensel lifting (Section 44.2 and 44.3), and presenting the general structure of the Kronecker package (Section 44.4) I deeply discuss its three steps (Sections 44.5, 44.6 and 44.7), its genericity conditions showing that the ’good’ choices live in an open Zariski set (Section 44.8) and sketch its complexity analysis (Sections 44.9).

44.1 Kronecker parametrization

Let

I ⊂ k[X_{1}, …, X_{n}] be an unmixed radical ideal, d := dim(I), r := n − d = r(I) the dimension and the rank of I; M := (c_{ij}) ∈ GL(n, k) be an invertible n × n square matrices with entries in

k such that, denoting Y_{i} := ∑_{j} c_{ij}X_{j} for each i,

{Y_{1}, …, Y_{n}} = {V_{1}, …, V_{d}, Z_{1}, …, Z_{r}}

is a Noether position (Definition 27.9.4) for I; K := k(V_{1}, …, V_{d}), and K ⊂ Ω(k) its algebraic closure; J := IK[Z_{1}, …, Z_{r}], the 0-dimesional extension of I; s := deg(J) the multiplicity (Definition 27.12.9 and 27.13.7) of I.

Recall (Section 34.2) that a K-linear form

U := λ_{1}Z_{1} + ⋯ + λ_{r}Z_{r}, λ_{i} ∈ K, λ_{1} ≠ 0,

is a primitive element of K[Z_{1}, …, Z_{r}]/J iff

Span_{K}{1, U, U^{2}, …, U^{s−1}} ≅ K[Z_{1}, …, Z_{r}]/J.

Definition 44.1.1 (Giusti–Heintz–Morales–Pardo). With the notation above, the assignment of

a matrix M := (c_{ij}) ∈ GL(n, k) such that, setting Y_{i} := ∑_{j} c_{ij}X_{j},

{Y_{1}, …, Y_{n}} = {V_{1}, …, V_{d}, Z_{1}, …, Z_{r}}

is a Noether position for I; a primitive element

U := λ_{1}Z_{1} + ⋯ + λ_{r}Z_{r}, λ_{i} ∈ K, λ_{1} ≠ 0,

of K[Z_{1}, …, Z_{r}]/J; the minimal polynomial q(T) := g_{0}(T) ∈ k[V_{1}, …, V_{d}][T] of U; the parametrization (g_{1}(T), …, g_{r}(T)), g_{i} ∈ k(V_{1}, …, V_{d})[T] of the variety

Z(I) such that Z_{i} − g_{i}(U) ∈ J for each i,

is called a geometric resolution of the variety Z(I).

Remark 44.1.2 (Giusti–Lecerf–Salvy).

Recalling Kronecker’s result (41.3) and Proposition 42.9.3, one can remark that for each polynomial p(T) ∈ K[T] which is relatively prime with q(T), and thus invertible in K[T]/q(T), one obtains, setting

w_{i}(T) := Rem(p(T)g_{i}(T), g_{0}(T)) ∈ k(V_{1}, …, V_{d})[T], 1 ≤ i ≤ r,

another parametrization (\frac{w_{1}(T)}{p(T)}, …, \frac{w_{r}(T)}{p(T)}) of the variety Z(I), with

p(U)Z_{i} − w_{i}(U) ∈ J for each i.

In particular the results (41.3) by Kronecker (where g_{0} is assumed irreducible) and of Proposition 42.9.3 (where g_{0} is assumed squarefree) are

obtained setting p := \frac{∂q}{∂T}.  �

Definition 44.1.3 (Giusti–Lecerf–Salvy). A parametrization

\{ q(V_{1}, …, V_{d}, T) = 0, ; \frac{∂q}{∂T} (V_{1}, …, V_{d}, T)Z_{1} = w_{1}(V_{1}, …, V_{d}, T) ; ⋮ ; \frac{∂q}{∂T} (V_{1}, …, V_{d}, T)Z_{r} = w_{r}(V_{1}, …, V_{d}, T)

of a radical and equidimensional ideal I ⊂ P, dim(I) = d, in ’generic’ position

is called a Kronecker parametrization of I.  �

The discussion above allows to state

Proposition 44.1.4 (Giusti–Lecerf–Salvy). With the notation above one can wlog assume

(1) q(T) := g_{0}(T) ∈ k[V_{1}, …, V_{d}][T];

(2) for each i ≤ r, w_{i} ∈ k[V_{1}, …, V_{d}][T];

(3) for each i ≤ r, \frac{∂q}{∂T} g_{i}(T) ≡ w_{i}(T) mod q(T),

(4) q(U) ∈ I,

(5) for each i ≤ r, \frac{∂q}{∂T} (U)Z_{i} − w_{i}(U) ∈ I;

(6) deg(q) = deg_{T} (q) = deg(J),

(7) deg(w_{i}) < deg_{T} (q) = deg(J) for each i ≤ r.

Moreover, for any radical ideal I, given any system of coordinate

{Y_{1}, …, Y_{n}} = {V_{1}, …, V_{d}, Z_{1}, …, Z_{r}}

which is in Noether position for I and any primitive element, there is a unique geometric resolution of the variety Z(I).

Proof. To prove that, we consider (compare Sections 41.8 and 41.9) new varables Λ_{d+1}, …, Λ_{n}, the field K_{Λ} := K(Λ_{d+1}, …, Λ_{n}), the extension I_{Λ} := IK_{Λ}[Z_{1}, …, Z_{r}] of I in K_{Λ}[Z_{1}, …, Z_{r}], the K(Λ_{d+1}, …, Λ_{n})-linear form

U_{Λ} := Λ_{d+1}Z_{1} + ⋯ + Λ_{n}Z_{r}

which is a primitive element of I_{Λ}, and its characteristic polynomial q_{Λ}(T) ∈ (K_{Λ}[Z_{1}, …, Z_{r}]/I_{Λ}) [T], which is squarefree, monic and of degree deg(I^{e}).

Differentiating q_{Λ}(T) with respect to each Λ_{d+i} we deduce the geometric resulution

\{ q_{Λ}(V_{1}, …, V_{d}, T) = 0, ; \frac{∂q_{Λ}}{∂T} (V_{1}, …, V_{d}, T)Z_{1} = −\frac{∂q_{Λ}}{∂Λ_{d+1}} (V_{1}, …, V_{d}, T) ; ⋮ ; \frac{∂q_{Λ}}{∂T} (V_{1}, …, V_{d}, T)Z_{r} = −\frac{∂q_{Λ}}{∂Λ_{n}} (V_{1}, …, V_{d}, T)

�

On the basis of these considerations, the TERA group aimed to solve the following

Problem 44.1.5 (Giusti–Heintz–Morales–Pardo). Let

f_{1}, …, f_{r}, g ∈ K[Z_{1}, …, Z_{r}]

and denote, for each ρ,

Z_{ρ} := {α ∈ K^{r} : f_{1}(α) = ⋯ = f_{ρ}(α) = 0 ≠ g(α)} J_{ρ} := 𝕀(f_{1}, …, f_{ρ}) : g^{∞}, L_{ρ} := \sqrt{J_{ρ}}, V_{ρ} := Z(J_{ρ})

so that

I(Z_{ρ}) = I(V_{ρ}) = L_{ρ}, V_{ρ} = Z(L_{ρ}) = ZI(Z_{ρ}).

Assuming that

(1) Z_{r} is finite;

(2) for each ρ, dim(L_{ρ}) = r − ρ;

(3) the Jacobian matrix (\frac{∂f_{i}}{∂Z_{j}} ) of f_{1}, …, f_{ρ} w.r.t. Z_{1}, …, Z_{r} has rank ρ at each point of V_{ρ}, compute a parametrization

{ q(U) = 0, ; Z_{1} = w_{1}(U) ; ⋮ ; Z_{r} = w_{r}(U)

of Z_{r}, where q(U) ∈ K[U] and w_{i}(U) ∈ K(U) for each i.  �

Remark 44.1.6. This problem can be easily justified via the considerations of Remark 35.3.9, Theorem 35.6.8 and Remark 35.6.9 on the ARGH-scheme; at each step of computation one obtains an ideal 𝔣 ⊂ ℚ[X_{1}, …, X_{n}] and a polynomial g ∈ ℚ[X_{1}, …, X_{n}] and one needs to compute the roots of the ideal

(\sqrt{𝔣 : g^{∞}})^{e} = \sqrt{𝔣^{e} : g^{∞}} ⊂ ℚ(V_{1}, …, V_{d})[Z_{1}, …, Z_{r}]

where we wlog assume that {V_{1}, …, V_{d}, Z_{1}, …, Z_{r}} is in Noether position for 𝔣 and d = dim(𝔣).

Denoting (g_{1}, …, g_{s}) any basis of 𝔣^{e} it is sufficient (cf. Corollary 36.1.6) to perform a generic linear combination f_{i} := ∑_{j=1}^{s} λ_{ij}g_{j} to obtain a regular sequence f_{1}, …, f_{r}.

Thus the setting related to the ARGH-scheme coincides with the one of Problem 44.1.5; such problem thus can be interpreted as solving an ARGHcomponent of a given ideal by producing its Kronecker parametrization.

Moreover, in this setting

(1) 𝔣^{e} is zero-dimensional;

(2) each L_{ρ} has rank ρ since f_{1}, …, f_{r} is a regular sequence;

(3) it has been proved^{3} that the Jacobian condition is satisfied by any generic

combination of the basis elements. �

44.2 Lifting Points

The variety Z(f_{1}, …, f_{r}) is a subvariety of the δ-dimensional variety, V_{ρ},

V_{ρ} ⊃ Z(f_{1}, …, f_{r}), δ = r − ρ,

defined by the polynomials f_{1}, …, f_{ρ} which satisfy conditions (2-3) of Problem 44.1.5. We call this sequence of polynomials a lifting system of V_{ρ}. Let us now consider a new system of coordinates {Y_{1}, …, Y_{r}} which is a Noether position for L_{ρ} and the projection ϕ : K^{r} ↦ K^{δ} defined by ϕ(a_{1}, …, a_{r}) = (a_{1}, …, a_{δ}).

3 Krick T., Pardo L.M., Une approache informatique pour l’approximation diophantienne, C.R. Acad. Sci. Paris 318 (1994), 407–412.

Definition 44.2.1. A point p := (p_{1}, …, p_{δ}) ∈ K^{δ} is a called a lifting point of V_{ρ} w.r.t. the lifting system f_{1}, …, f_{ρ} (and the frame {Y_{1}, …, Y_{r}}) if the Jacobian matrix of f_{1}, …, f_{ρ} w.r.t. Y_{δ+1}, …, Y_{r} is invertible at each point of the variety V_{p} := V_{ρ} ∩ ϕ^{−1}(p_{1}, …, p_{δ}) whose 0-dimensional ideal I(V_{p}) we denote L_{p}.

Definition 44.2.2. With the notation above, the assignment of

a lifting system f_{1}, …, f_{ρ} of V_{ρ} a matrix M := (c_{ij}) ∈ GL(r, K) such that {Y_{1}, …, Y_{r}}, Y_{i} := ∑_{j} c_{ij}Z_{j}, is

a Noether position for V_{ρ}; a lifting point p := (p_{1}, …, p_{δ}) of V_{ρ} w.r.t. the lifting system f_{1}, …, f_{ρ} and

the frame {Y_{1}, …, Y_{r}}; a primitive element U := λ_{δ+1}Y_{δ+1} + ⋯ + λ_{r}Y_{r}, λ_{i} ∈ K, λ_{δ+1} ≠ 0, of

K[Y_{δ+1}, …, Y_{r}]/L_{p}; the minimal polynomial q(T) of U; the parametrization (g_{δ+1}(T), …, g_{r}(T)) of V_{p}

so that

• Y_{j} − g_{j}(U) ∈ L_{p} for each j, δ < j ≤ r,

• V_{p} = {(p_{1}, …, p_{δ}, g_{δ+1}(α), …, g_{r}(α)) : α ∈ R}, R := {α ∈ K : q(α) = 0}

is called a lifting fiber of V_{ρ}.

Remark 44.2.3. Denoting

M^{−1} := (d_{ij}) ∈ GL(r, K) the inverse of M, h_{i}(Y_{1}, …, Y_{r}) := f_{i}(∑_{j} d_{1j}Y_{j}, …, ∑_{j} d_{rj}Y_{j}),

the following relations are satisfied by the data above:

(1) U(g_{δ+1}(T), …, g_{r}(T)) = λ_{δ+1}g_{δ+1}(T) + ⋯ + λ_{r}g_{r}(T) = T,

(2) h_{i}((p_{1}, …, p_{δ}, g_{δ+1}(T), …, g_{r}(T)) ∈ 𝕀(q(U)),

(3) s := deg(L_{p}) = deg(L_{r}).

Moreover, by Proposition 44.1.4, for any lifting fiber of V_{ρ}, there exists a unique geometric resolution of V_{ρ} for the same Noether position and primitive

element.  �

Proposition 44.2.4. The specialization of the minimal polynomial and the parametrization of this geometric resolution on the lifting point p gives exactly the minimal polynomial and the parametrization of the lifting fiber.

Proof. Assume that U is not a primitive element of V_{ρ}; we can then choose a primitive element U′ of V_{ρ} which is also a primitive element for V_{p} too. The specialization of the corresponding Kronecker parametrization of V_{ρ} gives a parametrization of V_{p}. By linear algebra on

Span_{K}{1, U′, U^{′2}, …, U^{′s−1}} ≡ K[Y_{δ+1}, …, Y_{r}]/L_{p}

one can compute the minimal polynomial of U whose degree is necessarily

less than s giving the required contradiction.  �

Lemma 44.2.5. With the notation and assumptions above, the set of points (p_{1}, …, p_{δ}, λ_{δ+1}, …, λ_{r}) ∈ K^{r} such that either

p := (p_{1}, …, p_{δ}) is not a lifting point or U := λ_{δ+1}Y_{δ+1} + ⋯ + λ_{r}Y_{r} is not a primitive element for V_{p}

are contained into an algebraic proper subset of K^{r}.

Proof. Let J the Jacobian matrix of f_{1}, …, f_{ρ} w.r.t. Y_{δ+1}, …, Y_{r}.

The integral dependency relation of det(J) modulo L_{ρ} is given by a monic polynomial F (U) ∈ K[Y_{1}, …, Y_{δ}][U].

By assumption, det(J) is not a zero-divisor of K[Y_{1}, …, Y_{r}]/L_{ρ}, so that A(Y_{1}, …, Y_{δ}) := F (0) ≠ 0 satisfies A ∈ L_{ρ} + (det(J)) .

Each point p := (p_{1}, …, p_{δ}) such that A(p_{1}, …, p_{δ}) ≠ 0 is a lifting point. Let now fix a lifting point p := (p_{1}, …, p_{δ}) and consider (Compare Proposition 44.1.4)

• the ideal L_{Λ} := L_{p}K(Λ_{δ+1}, …, Λ_{r})[Y_{δ+1}, …, Y_{r}],

• Y_{Λ} := Λ_{δ+1}Y_{δ+1} + ⋯ + Λ_{r}Y_{r} ∈ K(Λ_{δ+1}, …, Λ_{n})[Y_{δ+1}, …, Y_{n}]/L

• q_{Λ}(T) ∈ (K(Λ_{δ+1}, …, Λ_{r})[Y_{δ+1}, …, Y_{r}]/L_{Λ}) [T] its minimal polynomial,

• Disc(q) ∈ K(Λ_{δ+1}, …, Λ_{r}) its discriminant (cf. Theorem 10.6.5).

Then any point (λ_{δ+1}, …, λ_{r}) such that Disc(q)(λ_{δ+1}, …, λ_{r}) ≠ 0 gives a

primitive element Y := λ_{δ+1}Y_{δ+1} + ⋯ + λ_{r}Y_{r} for V_{p}.  �

44.3 Newton–Hensel Lifting

Proposition 44.3.1. Let

R be an integral domain, I an ideal of R, I^{⋆} ⊂ R[T] its extension, f := (f_{1}, …, f_{r}), f_{i} ∈ R[Z_{1}, …, Z_{r}], U := λ_{1}Z_{1} + … λ_{r}Z_{r}, λ_{i} ∈ R a linear form, q(T) ∈ R[T] a monic polynomial, s := deg(q) > 1, v := (v_{1}(T), …, v_{r}(T)), v_{i}(T) ∈ R[T], deg(v_{i}) < s, J = (\frac{∂f_{i}}{∂Z_{j}} ) the Jacobian matrix of f_{1}, …, f_{r} w.r.t. Z_{1}, …, Z_{r}

and assume that the following relations

(A) f_{j}(v_{1}(T), …, v_{r}(T)) ≡ 0 mod I^{⋆} + (q), for each j, (B) T ≡ λ_{1}v_{1}(T) + … λ_{r}v_{r}(T) mod I^{⋆} + (q), (C) J(v_{1}(T), …, v_{r}(T)) is invertible modulo I^{⋆} + (q),

hold in R[T]; then the following objects exist and can be computed:

• a monic polynomial Q(T) ∈ R[T],

• V := (V_{1}(T), …, V_{r}(T)), V_{i}(T) ∈ R[T],

such that in R[T] hold

(1) deg(Q) = s,

(2) Q(T) ≡ q(T) mod I^{⋆},

(3) deg(V_{i}) < s, for each i,

(4) V_{i}(T) ≡ v_{i}(T) mod I^{⋆}, for each i,

(5) f_{j}(V_{1}(T), …, V_{r}(T)) ≡ 0 mod (I^{⋆})^{2} + (Q), for each j,

(6) T ≡ λ_{1}V_{1}(T) + … λ_{r}V_{r}(T) mod (I^{⋆})^{2} + (Q).

Proof. Consider a generic vector

w := (w_{1}(T), …, w_{r}(T)), w_{i}(T) ∈ R[T], deg(w_{i}) < s

and write the Taylor expansion of f between w and v:

f(w) = f(v) + J(v) · (w − v) + ⋯

Since the aim is to find a such vector w which moreover satisfies

w ≡ v mod I^{⋆} and f(w) ≡ 0 mod (I^{⋆})^{2} + (q),

we use such conditions on the expension above obtaining

0 ≡ f(w) ≡ f(v) + J(v) · (w − v) mod (I^{⋆})^{2} + (q)

thus deducing, thanks of assumption (C), the existence and the uniqueness of the solution

w := v − J^{−1}(v) · f(v) mod (I^{⋆})^{2} + (q).

The polynomial

Δ(T) := U(w) − T = λ_{1}w_{1}(T) + … λ_{r}w_{r}(T) − T ∈ R[T],

is such that deg(Δ) < s and assumptions (B-C) allow to deduce that all his coefficients are member of I.

Setting Y := T + Δ(T) we have

Δ(Y) = Δ(T) + Δ′(T)(Y − T) + ⋯ = Δ(T) + Δ′(T)Δ(T) + ⋯

hence Δ(Y) ≡ Δ(T) = Y − T mod I^{2}.

Therefore defining p(T), u_{i}(T) as the unique polynomials such that

p − q′Δ, u_{i} − w′_{i}Δ ∈ (q), deg(p) < s, deg(u_{i}) < s,

we have, mod(I^{⋆})^{2} + (q),

Q(Y) := q(Y) − p(Y) ≡ q(Y) − Δ(Y)q′(Y) ≡ q(T) − (Y − T)q′(T) ≡ q(T) and

V_{i}(Y) := w_{i}(Y) − u_{i}(Y) ≡ w_{i}(Y) − Δ(Y)w′(Y) ≡ w_{i}(T) − (Y − T)w′(T) ≡ w_{i}(T)

We have therefore the ideal equality

H := (q(T), Y − T − Δ(T), Z_{1} − w_{1}(T), …, Z_{r} − w_{r}(T))
= (Q(Y), T − Y − Δ(Y), Z_{1} − V_{1}(Y), …, Z_{r} − V_{r}(Y))

in the ring (R/I^{2}) [U, Y, Z_{1}, …, Z_{r}].

Therefore Q and V satisfy the required conditions:

(1) deg(p) < s = deg(q) ⟹ deg(Q) = s;

(2) it is sufficient to remark that p ∈ I;

(3) deg(u_{i}) < s, deg(w_{i}) < s ⟹ deg(V_{i}) < s, for each i;

(4) it is sufficient to remark that u_{i} ∈ I.

(5) We have

f_{j}(V_{1}(Y), …, V_{r}(Y)) ≡ f_{j}(Z_{1}, …, Z_{r}) ≡ f_{j}(w_{1}(T), …, w_{r}(T)) mod H

and f_{j}(w_{1}(T), …, w_{r}(T)) ∈ (I^{⋆})^{2} + (q) hence

f_{j}(V_{1}(Y), …, V_{r}(Y)) ∈ ((I^{⋆})^{2} + L) ∩ R[Y] = (I^{⋆})^{2} + (Q(Y)).

(6) Since, mod(I^{⋆})^{2} + H,

Y − U(V_{1}(Y), …, V_{r}(Y)) ≡ Y − U(Z_{1}, …, Z_{r})
≡ Y − U(w_{1}(T), …, w_{r}(T))
= Y − T − Δ(T)
= 0

we have Y −U(V_{1}(Y), …, V_{r}(Y)) ∈ ((I^{⋆})^{2} + L)∩R[Y] = (I^{⋆})^{2} +(Q(Y)).

�

Corollary 44.3.2. With the same notation and assumptions as in Proposition 44.3.1 for any ϵ ∈ ℕ the following objects exist and can be computed

• a monic polynomial Q(T) ∈ R[T],

• V := (V_{1}(T), …, V_{r}(T)), V_{i}(T) ∈ R[T],

such that

(1) deg(Q) = s,

(2) Q(T) ≡ q(T) mod I^{⋆}

(3) deg(V_{i}) < s, for each i,

(4) V_{i}(T) ≡ v_{i}(T) mod I^{⋆}, for each i,

(5) f_{j}(V_{1}(T), …, V_{r}(T)) ≡ 0 mod (I^{⋆})^{ϵ+1} + (Q), for each j,

(6) T ≡ λ_{1}V_{1}(T) + … λ_{r}V_{r}(T) mod (I^{⋆})^{ϵ+1} + (Q)

hold in R[T].  �

44.4 Kronecker Package: Description

The solution of Problem 44.1.5 is incremental on the number of equations to be solved, thus iteratively solving each system V_{ρ}, each resolution being encoded by means of a lifting fiber.

Thus, at each step the algorithm depends on the choice of a Noether position for V_{ρ}, a lifting point and a primitive element, Zariski-openness granting that such choices can be done randomly.

Let us therefore assume we have

the δ = r − ρ-dimensional variety V_{ρ} ⊂ Z(f_{1}, …, f_{r}), the projection ϕ : K^{r} ↦ K^{δ} defined by ϕ(a_{1}, …, a_{r}) = (a_{1}, …, a_{δ}), the lifting system f_{1}, …, f_{ρ} of V_{ρ}, a frame of coordinates which is in Noether position for V_{ρ} and which, by

simplicity, we assume to be {Z_{1}, …, Z_{r}}, a lifting point p := (p_{1}, …, p_{δ}) of V_{ρ} w.r.t. the lifting system f_{1}, …, f_{ρ} and

the frame {Z_{1}, …, Z_{r}}, the primitive element U := λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r}, λ_{i} ∈ k, λ_{δ+1} ≠ 0, of

K[Z_{δ+1}, …, Z_{r}]/L_{p}, the minimal polynomial q(T) of U, the parametrization (v_{δ+1}(T), v_{δ+2}(T), …, v_{r}(T)) of both V_{p} and V_{ρ}.

Up to now we simply assume that

• the Noether position {Z_{1}, …, Z_{r}},

• the lifting point p := (p_{1}, …, p_{δ}), and

• the primitive element λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r}

are sufficiently generic in order to satisfy all the conditions of genericity required by the algorithm; we will discuss deepler such conditions in Section 44.8.

\underline{Lifting Step} Thus we are assuming to have a geometric resolution

{ q(T) = 0, ; Z_{δ+1} = v_{δ+1}(T) ; ⋮ ; Z_{r} = v_{r}(T)

for the primitive element

U := λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r} ∈ K[Z_{δ+1}, …, Z_{r}]/L_{p}

of the variety V_{p} defined by

a := (p_{1}, …, p_{δ}, α_{δ+1}, …, α_{r}) ∈ V_{p} ⟺ f_{1}(a) = … = f_{ρ}(a) = 0 ≠ g(a)

and the 0-dimensional radical ideal

L_{p} := I(V_{p}) = L_{ρ} + (Z_{1} − p_{1}, …, Z_{δ} − p_{δ})

and we compute a geometric resolution

{ Q(Z_{δ}, T) = 0, ; Z_{δ+1} = V_{δ+1}(Z_{δ}, T) ; ⋮ ; Z_{r} = V_{r}(Z_{δ}, T)

for the primitive element

U := ∑_{i=1}^{ρ} λ_{δ+i}Z_{δ+i} ∈ k(Z_{δ})[Z_{δ+1}, …, Z_{r}]/L_{D}^{e}

of the variety V_{D} defined by a = (p_{1}, …, p_{δ−1}, α_{δ}, …, α_{r}) ∈ V_{D} ⟺ f_{1}(a) = ⋯ = f_{ρ}(a) = 0 ≠ g(a) and the 1-dimensional radical ideal

L_{D} := I(V_{D}) = L_{ρ} + (Z_{1} − p_{1}, …, Z_{δ−1} − p_{δ−1}).

\underline{Intersection Step} From this date we compute a geometric resolution

{ q(Z) = 0, ; Z_{δ} = v_{δ}(Z) ; ⋮ ; Z_{r} = v_{r}(Z)

for the primitive element

U := ∑_{j=0}^{ρ} λ_{δ+j}Z_{δ+j} ∈ K[Z_{δ}, …, Z_{r}]/L′

of the 0-dimensional radical ideal

L′ := \sqrt{L_{r} + (Z_{1} − p_{1}, …, Z_{δ−1} − p_{δ−1}, f_{ρ+1})}.

\underline{Cleaning Step} We now remove the points a ∈ Z(L′) such that g(a) = 0 thus getting the required geometric resolution

{ q′(T) = 0, ; Z_{δ} = v′_{δ}(T) ; ⋮ ; Z_{r} = v′_{r}(T)

for the primitive element

236 44. Kronecker IV

U := λ_{δ}Z_{δ} + ⋯ + λ_{r}Z_{r} ∈ K[Z_{δ}, …, Z_{r}]/L_{p′}

and the lifting point p′ := (p_{1}, …, p_{δ−1}) of the variety V_{p′} defined by

a := (p_{1}, …, p_{δ−1}, α_{δ}, …, α_{r}) ∈ V_{p′} ⟺ f_{1}(a) = … = f_{ρ+1}(a) = 0 ≠ g(a)

and the 0-dimensional radical ideal L_{p′} := I(V_{p}) = L_{ρ+1} + (Z_{1} − p_{1}, …, Z_{δ−1} − p_{δ−1}).

44.5 Kronecker Package: Lifting Step

Defining ϵ := deg(L_{p}) + 1 = deg(L_{ρ}) + 1 h_{i}(Z_{δ}, Z_{δ+1}, …, Z_{r}) := f_{i}(p_{1}, …, p_{δ−1}, Z_{δ}, Z_{δ+1}, …, Z_{r}) we can apply Corollary 44.3.2 to the data R := K[Z_{δ}], I := 𝕀(Z_{δ}) ⊂ R, I^{⋆} := 𝕀(Z_{δ}) ⊂ K[Z_{δ}, T], f := (h_{1}, …, h_{r}), h_{i} ∈ R[Z_{δ+1}, …, Z_{r}], U := λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r}, q(T) ∈ R[T] the minimal polynomial of U, v := (v_{δ+1}(T), v_{δ+2}(t), …, v_{r}(T)), thus obtaing a monic polynomial Q(T) ∈ K[Z_{δ}][T], V := (V_{δ+1}(T), …, V_{r}(T)), V_{i}(T) ∈ K[Z_{δ}][T], such that

(1) h_{j}(V_{δ+1}(T), …, V_{r}(T)) ≡ 0 mod I^{ϵ+1} + (Q), for each j,

(2) T ≡ λ_{δ+1}V_{1}(T) + … λ_{r}V_{r}(T) mod I^{ϵ+1} + (Q) id est a polynomial Q(Z_{δ}, T) ∈ K[Z_{δ}, T], V := (V_{δ+1}(Z_{δ}, T), …, V_{r}(Z_{δ}, T)), V_{i}(Z_{δ}, T) ∈ K[Z_{δ}, T], such that

(1) f_{j}(p_{1}, …, p_{δ−1}, Z_{δ}, V_{δ+1}(Z_{δ}, T), …, V_{r}(Z_{δ}, T)) ≡ 0 mod Q, for each j,

(2) T ≡ λ_{δ+1}V_{δ+1}(Z_{δ}, T) + … λ_{r}V_{r}(Z_{δ}, T) mod Q.

We therefore have the parametrization

{ Q(Z_{δ}, T) = 0, ; Z_{δ+1} = V_{δ+1}(Z_{δ}, T) ; ⋮ ; Z_{r} = V_{r}(Z_{δ}, T)

of

V_{D} := {a = (p_{1}, …, p_{δ−1}, α_{δ}, …, α_{r}) ∈ K^{r} : f_{1}(a) = ⋯ = f_{ρ}(a) = 0 ≠ g(a)}

and the 1-dimensional radical ideal

L_{D} := I(V_{D}) = L_{ρ} + (Z_{1} − p_{1}, …, Z_{δ−1} − p_{δ−1}).

44.6 Kronecker Package: Intersection Step

Let us therefore assume to have a 1-dimensional ideal

I ⊂ K[Z_{δ}, Z_{δ+1}, …, Z_{r}]

and its radical L := \sqrt{I} given by means of a geometric resolution

{ Q(Z_{δ}, T) = 0, ; Z_{δ+1} = V_{δ+1}(Z_{δ}, T) ; ⋮ ; Z_{r} = V_{r}(Z_{δ}, T)

for

• the Noether position{Z_{δ}, Z_{δ+1}, …, Z_{r}} for L and

• the primitive element U := λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r}, λ_{i} ∈ K ⊂ K(Z_{δ}),

and a polynomial f ∈ K[Z_{δ}, Z_{δ+1}, …, Z_{r}] such that L+(f) is 0-dimensional^{4}.

Let us consider Q(Z_{δ}, T) and f(Z_{δ}, V_{δ+1}(Z_{δ}, T), …, V_{r}(Z_{δ}, T)) as polynomials in K[Z_{δ}][T] and their resultant:

Lemma 44.6.1. Denoting

• A(Z_{δ}) := Res(Q, f(Z_{δ}, V_{δ+1}, …, V_{r}) ∈ K[Z_{δ}],

• B := K[Z_{δ}, Z_{δ+1}, …, Z_{r}]/L,

• B′ := K(Z_{δ})[Z_{δ+1}, …, Z_{r}]/L^{e},

• F (T) ∈ K[Z_{δ}][T] the integral dependency relation of f modulo L,

4 We apply the results of this section to the case L := L_{D} and

f(Z_{δ}, Z_{δ+1}, …, Z_{r}) := f_{ρ+1}(p_{1}, …, p_{δ−1}, Z_{δ}, Z_{δ+1}, …, Z_{r}).

• Φ_{f} the endomorphism of multiplication by f in B′,

• χ_{f}(T) ∈ K(Z_{δ})[T] the characteristic polynomial of Φ_{f},

• m_{f}(T) ∈ K(Z_{δ})[T] the monic polynomial of Φ_{f},

• W := Z(L^{e}) = {b_{1}, … b_{t}}, σ_{i} := mult(b_{i}, L^{e}),

• ϕ : K^{ρ+1} ↦ K the projection ϕ(α, β_{1}, …, β_{ρ}) = α,

• for each α ∈ K, W_{α} := {a_{1}, … a_{s}} = π^{−1}(α) ∩ Z(L), each a_{i} being counted with the proper multiplication s_{i} := mult(a_{i}, L), ∑_{i} s_{i} = deg(L^{e}),

we have

(1) m_{f}(T), χ_{f}(T) ∈ K[Z_{δ}][T];

(2) setting χ_{f} := ∑_{i} C_{i}(Z_{δ})T^{i} and s := deg(L^{e}) we have C_{i}(Z_{δ}) ∈ K[Z_{δ}] and deg(C_{i}) ≤ (s − i) deg(f);

(3) C_{0}(Z_{δ}) ∈ L + (f);

(4) C_{0}(α) = ∏_{a_{i}∈W_{α}} f(a_{i})^{s_{i}}.

(5) C_{0}(Z_{δ}) and A(Z_{δ}) concide up to the sign;

(6) deg(A) ≤ s deg(f);

(7) {α ∈ K : A(α) = 0} = {ϕ(b) : b ∈ W, f(b) = 0};

Proof. Since F (Φ_{f}) = 0 we deduce that m_{f} | F and, since both are monic, Gauss Lemma (Corollary 6.1.5) implies (1). We moreover know (Corollary 40.5.2) that χ(Z_{δ}, T) = ∏_{i=1}^{t} (T − f(Z_{δ}, b_{i}))^{σ_{i}} so that we deduce (2).

Remark that B is a finite K[Z_{δ}]-module of rank s := deg(L^{e}).

Since any K[Z_{δ}]-basis of B induces a K(Z_{δ})-basis of B′ (cf. Section 36.3), the characteristic polynomial of Φ_{f} in B and B′ coincide, so that Cayley-Hamilton theorem in B implies χ(Z_{δ}, f) ∈ L and hence (3).

Moreover, for each α ∈ K, denoting B_{0} := K[Z_{δ}, Z_{δ+1}, …, Z_{r}]/L+(Z_{δ}−α) and remarking that the specalization at α of the K[Z_{δ}]-basis of B gives a K-basis of B_{0}, we deduce that C_{0}(α) is the constant coefficient of the characteristic polynomial

χ(T) = ∏_{i=1}^{s} (T − f(a_{i}))^{s_{i}}

of the multiplication by f in B_{0} whence (4).

Since

• a ∈ Z(L + (f)) ⟹ C_{0}(π(a)) = 0 for each a ∈ K^{ρ+1} as a consequence of (3) and,

• for each α ∈ K which annihilates C_{0}, (4) implies the existence of a ∈ π^{−1}(α) ∩ Z(L) which annihilates f(a),

we obtain (5) of which (6-7) are direct consequences.  �

Since probably Z_{δ} is not a primitive element for

K[Z_{δ}, Z_{δ+1}, …, Z_{r}]/\sqrt{L + (f)},

while this is true for a generic element λ_{δ}Z_{δ} + U, let us therefore introduce a new variable Z, denote

ˆ·: K[Z_{δ}, Z_{δ+1}, …, Z_{r}][T] ↦ K[Z, Z_{δ+1}, …, Z_{r}][T]

the substitution

ĝ := g(λ_{δ}^{−1}(Z − T), T, Z_{δ+1}, …, Z_{r}) for each g(Z_{δ}, T, Z_{δ+1}, …, Z_{r}),

and assume that Z = (λ_{δ}Z_{δ} + T)ˆ has the required properties, as it is true for almost choices of λ_{δ}.

Definition 44.6.2. A point λ_{δ} ∈ k is called a Liouville point w.r.t. the above geoemetric resolution of L if

(1) λ_{δ} ≠ 0

(2) Q̂ is monic in T and deg_{T} (Q̂) = deg_{T} (Q) = s = deg(L),

(3) Q̂ is squarefree and relatively prime with P̂, P := \frac{∂Q}{∂T}.

Lemma 44.6.3. With the above notation, if λ_{δ} is a Liouville point then the variables {Z, Z_{δ+1}, …, Z_{r}} are in Noether position w.r.t. L̂ := {f̂ : f ∈ L} and

{ Q̂(Z, T) = 0, ; Z_{δ+1} = V̂_{δ+1}(Z, T) ; ⋮ ; Z_{r} = V̂_{r}(Z, T)

is a geometric resolution of L̂ for the primitive element U.

Proof. First of all, L̂∩K[Z] = {0}, since for each h(Z) ∈ L̂∩K[Z], Q(Z_{δ}, T) | h(λ_{δ}Z_{δ} + T) and Q̂(Z, T) | h(Z); since Q̂(Z, T) is monic in T this implies h(Z) = 0 as required.

Thus, in order to prove that {Z, Z_{δ+1}, …, Z_{r}} is in Noether position it is sufficient to prove that each Z_{i} is dependent over Z: denote L_{1} := L̂ + (T − U) ⊂ K[Z, Z_{δ+1}, …, Z_{r}, T] and consider a bivariate polynomial h(Z_{δ}, Z_{i}) ∈ L, monic and whose total degree is bounded by deg_{Z_{i}}(h), whose existence is implied by the assumption that {Z_{δ}, Z_{δ+1}, …, Z_{r}} is in Noether position w.r.t. I; then h(λ_{δ}^{−1}(Z−T), Z_{i}) ∈ L and (since Q̂(Z, T) ∈ L and its total degree is bounded by s) we can deduce the existence of a polynomial H(Z, Z_{i}) ∈ L, monic and whose total degree is bounded by deg_{Z_{i}}(h), i.e. the dependency of Z_{i} over Z.

Since Q̂ remains squarefree, U remains primitive.  �

Lemma 44.6.4. Almost each λ_{δ} ∈ K is a Liouville point.

Proof. Setting W := λ_{δ}^{−1}Z we have

Q̂(Z, T) = Q(λ_{δ}^{−1}(Z − T), T) = Q(W − λ_{δ}^{−1}T, T).

Both the discriminant of Q(W − ΛT, T) and the resultant, in K[W, Λ][T] of Q(W − ΛT, T) with \frac{∂Q}{∂T}(W − ΛT, T)

are polynomials in K[W, Λ] and do not vanish for Λ = 0; hence almost all choice for λ_{δ} ≠ 0 grants (3). Also denoting h(Z, T) := H(Q) (Definition 23.2.1) the homogeneous part of maximal degree s of Q so that the coefficient of T^{s} in Q̂(Z, T) is h(−λ_{δ}^{−1}, 1); since again h(0, 1) ≠ 0, almost all

choice for λ_{δ} ≠ 0 grants (2).  �

Remark 44.6.5. If λ_{δ} ∈ K is a Liouville point w.r.t. the above geoemetric resolution of L, and f̂ denotes the polynomial f̂:= f(λ_{δ}^{−1}(Z −T), Z_{δ+1}, …, Z_{r}), the resultant A(Z) ∈ K[Z] of the polynomials (in K[Z][T]) Q̂(Z, T) and f(λ_{δ}^{−1}(Z − T), V̂_{δ+1}(Z, T), …, V̂_{r}(Z, T)) satisfies A(λ_{δ}Z_{δ} + U) ∈ L: in fact we already proved (Lemma 44.6.1) that A(Z) ∈ L̂ + (f̂); thus replacing Z with λ_{δ}Z_{δ} + U we obtain A(λ_{δ}Z_{δ} + U) ∈ L.

Moreover each root (α, β_{δ+1}, …, β_{r}) ∈ π^{−1}(α) of L̂, where A(α) = 0, corresponds to the root (β_{δ}, β_{δ+1}, …, β_{r}) of L + (f) where

β_{δ} = λ_{δ}^{−1} (α − ∑_{j=1}^{n} λ_{δ+j}β_{δ+j}) ,

or equivalently, α = ∑_{j=0}^{ρ} λ_{δ+j}β_{δ+j}.

This is not yet sufficient to describe Z(L + (f)) because we still miss the parametrization of the coordinates. Denoting

T_{δ}, T_{δ+1}, …, T_{r} new variables, K_{t} := K(T_{δ}, T_{δ+1}, …, T_{r}), L_{t} := LK_{t}[Z_{δ}, Z_{δ+1}, …, Z_{r}], U_{t} := U + T_{δ+1}Z_{δ+1} + ⋯ + T_{r}Z_{r} = ∑_{j=1}^{ρ}(λ_{δ+j} + T_{δ+j})Z_{δ+j},

let us assume to have the geometric resolution

{ q_{t}(Z_{δ}, T) = 0, ; Z_{δ+1} = V_{t,δ+1}(Z_{δ}, T) ; ⋮ ; Z_{r} = V_{t,r}(Z_{δ}, T)

of L_{t} for the primitive element U_{t}. Since, for a Liouville point λ_{δ} for L, λ_{δ} +T_{δ} is a a Liouville point for L_{t}, in this setting the resultant computation returns a polynomial A_{t}(Z) ⊂ K_{t}[Z] such that A_{t}((λ_{δ} + T_{δ})Z_{δ} + T) ∈ L_{t}; more precisely, if we express A_{t}(Z) as

A_{t}(Z) = A(Z) + T_{δ}A_{δ}(Z) + T_{δ+1}A_{δ+1}(Z) + ⋯ + T_{r}A_{r}(Z) + B

where A_{i}(Z) ∈ K[Z] and B(T_{δ}, T_{δ+1}, …, T_{r}, Z) ∈ (T_{δ}, T_{δ+1}, …, T_{r})^{2} and we evaluate it in

(λ_{δ} + T_{δ})Z_{δ} + U_{t} = (λ_{δ}Z_{δ} + U) + Z_{δ}T_{δ} + T_{δ+1}Z_{δ+1} + ⋯ + T_{r}Z_{r},

Taylor expansion allows to deduce that both A(λ_{δ}Z_{δ} + U) and

A′(λ_{δ}Z_{δ} + U) + Z_{δ+j}A_{δ+j}(λ_{δ}Z_{δ} + U), 0 ≤ j ≤ ρ

are members of L.

The roots of the polynomial A_{t}(Z) are the values of the linear form

U_{t} := ∑_{j=0}^{ρ} λ_{δ+j}(Z_{δ+j} + T_{δ+j})

at the roots of L + (f). Thus denoting Z(L + (f)) =: {a_{1}, … a_{s}}, and U the linear form U := ∑_{j=0}^{ρ} λ_{δ+j}Z_{δ+j} we have A_{t}(Z) = ∏_{j=1}^{s} (Z − U_{t}(a_{j}))^{s_{j}} ∈ L_{t} and

∏_{j=1}^{s}(Z − U_{t}(a_{j}))^{s_{j}} ≡ A(Z) + ∑_{i=0}^{r} A_{δ+i}(Z)T_{δ+i} mod L_{t} + (T_{δ}, T_{δ+1}, …, T_{r})^{2}.

Thus, by expansion, we obtain

A(Z) := ∏_{j=1}^{s}(Z − U(a_{j}))^{s_{j}}

A_{δ+i}(Z) := − ∑_{h=1}^{s} Z_{δ+i}(a_{h})s_{h}(Z − U(a_{h}))^{s_{h}−1} ∏_{j=1,j≠h}^{s}(Z − U(a_{j}))^{s_{j}}.

Thus D(Z) := gcd(A, A′) = ∏_{j=1,}^{s}(Z − U(a_{j}))^{s_{j}−1} divides each A_{δ+i} so that

{ A(Z)/D(Z) = 0, ; A′(Z)/D(Z)Z_{δ} = A_{δ}(Z)/D(Z) ; A′(Z)/D(Z)Z_{δ+1} = A_{δ+1}(Z)/D(Z) ; ⋮ ; A′(Z)/D(Z)Z_{r} = A_{r}(Z)/D(Z)

is the required geometric resolution of L + (f) for the primitive element U := ∑_{j=0}^{ρ} λ_{δ+j}Z_{δ+j}.

Finally, euclidean arithmetic allows to compute the inverse of A′/D modulo A/D and gives a representation

{ q(Z) = 0, ; Z_{δ} = v_{δ}(Z) ; ⋮ ; Z_{r} = v_{r}(Z)

�

In order to successfully apply Remark 44.6.5, we need to compute the geometric resolution

{ q_{t}(Z_{δ}, T) = 0, ; Z_{δ+1} = V_{t,δ+1}(Z_{δ}, T) ; ⋮ ; Z_{r} = V_{t,r}(Z_{δ}, T)

of L_{t} for the primitive element U_{t} starting with our data:

• the parametrization

{ Q(Z_{δ}, T) = 0, ; Z_{δ+1} = V_{δ+1}(Z_{δ}, T) ; ⋮ ; Z_{r} = V_{r}(Z_{δ}, T)

of L := L_{D} and

• the primitive element U := λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r}.

Since in K_{t}[Z_{δ}, Z_{δ+1}, …, Z_{r}]/L_{t} we have Z_{i} ≡ V_{i}(Z_{δ}, U) mod L_{t}, we obtain

U_{t} ≡ U + T_{δ+1}V_{δ+1}(Z_{δ}, U) + ⋯ + T_{r}V_{r}(Z_{δ}, U)
≡ U + T_{δ+1}V_{δ+1}(Z_{δ}, U_{t}) + ⋯ + T_{r}V_{r}(Z_{δ}, U_{t})

modulo L_{t} + (T_{δ}, T_{δ+1}, …, T_{r})^{2}, whence

U ≡ U_{t} − T_{δ+1}V_{δ+1}(Z_{δ}, U_{t}) − ⋯ − T_{r}V_{r}(Z_{δ}, U_{t}).

Therefore replacing U in the parametrization and applying Taylor expansion we obtain

q_{t}(Z_{δ}, T) := Q(Z_{δ}, T) − \frac{∂Q}{∂T} ∑_{i=1}^{ρ} T_{δ+i}V_{δ+i}(Z_{δ}, T)
≡ Q(Z_{δ}, T) − T_{δ+1}W_{δ+1}(Z_{δ}, T) − ⋯ − T_{r}W_{r}(Z_{δ}, T)

modL_{t} + (T_{δ}, T_{δ+1}, …, T_{r})^{2} and

V_{t,δ+i}(Z_{δ}, T) := V_{δ+i}(Z_{δ}, T) − \frac{∂Q}{∂T} ∑_{i=1}^{ρ} T_{δ+i}V_{δ+i}(Z_{δ}, T)

modL_{t} + (q_{t}) + (T_{δ}, T_{δ+1}, …, T_{r})^{2}.

Algorithm 44.6.6. In conclusion we have to compute

• the data q_{t} and V_{t,δ+i} with the formulae above; • the resultant A_{t}(Z) of the polynomials q_{t}(λ_{δ}^{−1}(Z − T), T) and

f_{ρ+1}(p_{1}, …, p_{δ−1}, λ_{δ}^{−1}(Z − T), V̂_{t,δ+1}(Z_{δ}, T), …, V̂_{t,r}(Z_{δ}(Z_{δ}, T))

• and, by expansion, the data A, D, A_{δ}, …, A_{r}.

This computation, mainly the one of A_{t}(Z), for complexity reason (see the discussion in page 246) is not performed in K(Z_{δ})[T] but in

R[T], R := K[T_{δ}, …, T_{r}][Z_{δ}]/(Z_{δ} − p)^{ds+1}

where p ∈ k is a ‘generic’ value, d := deg(f), s := deg(Q) so that ds counts the roots, with multiplicity, of L + (f).

44.7 Kronecker Package:Cleaning Step

So now we have the geometric resolution

{ q(Z) = 0, ; Z_{δ} = v_{δ}(Z) ; ⋮ ; Z_{r} = v_{r}(Z)

of L + (f) and we need to remove from W := Z(L + (f)) =: {a_{1}, … a_{s}} the roots such that g(a_{i}) = 0.

This is easily performed by computing

G(Z) := g(p_{1}, …, p_{δ−1}, v_{δ}(Z), …, v_{r}(Z)), e(Z) := gcd(q, G), q′ := q/e, w′_{i} := v_{i} mod q.

We need however to be sure that the lifting point p := (p_{1}, …, p_{δ−1}) is not bad: it is sufficient to be sure that

p ∉ π(W′′ ∩ Z(g))

where we are denoting

W′ := {a ∈ W : g(a) ≠ 0}, W′′ := ZI(W′), π : K^{r} ↦ K^{δ−1} the projection π(α_{1}, …, α_{r}) = (α_{1}, …, α_{δ−1}).

A such point is called a cleaning point. Clearly almost all lifting points are cleaning points: the bad points are the projections of the intersection of a variety with dimension δ − 1 and the hypersuface g; such projection over K^{δ−1} has dimension δ − 2.

44.8 Genericity conditions

With the same notation and assumption as in Section 44.4 we are now discussing the genericity conditions; we therefore begin with

• a Noether position {Z_{1}, …, Z_{r}},

• a lifting point p := (p_{1}, …, p_{δ}),

• a primitive element U := λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r} of K[Z_{δ+1}, …, Z_{r}]/L_{p}.

for V_{ρ}.

The computation we sketched in Section 44.4 and discussed in the next Sections returns a lifting fiber of V_{ρ+1} for the lifting point (p_{1}, …, p_{δ−1}) and the primitive element λ_{δ}Z_{δ} + λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r} provided the following conditions hold:

• {Z_{1}, …, Z_{r}} is in Noether position for V_{ρ+1};

• p := (p_{1}, …, p_{δ−1}) is a lifting point for V_{ρ+1};

• λ_{δ} is a Liouville point for L_{D};

• λ_{δ}Z_{δ} + λ_{δ+1}Z_{δ+1} + ⋯ + λ_{r}Z_{r} is a primitive element for V_{ρ+1};

• p := (p_{1}, …, p_{δ−1}) is a cleaning point for W′′.

We need moreover three further assumptions:

(1) {Z_{0}, …, Z_{r}} is in Noether position for each homogeneous ideal ^{h}L_{ρ} ⊂ K[Z_{0}, …, Z_{r}];

(2) p_{δ} is lucky for the truncated computation;

(3) U and λ_{δ} are lucky for the resultant computation of Remark 44.6.5.

In fact

(1) consider, as an example, the 1-dimensional prime ideal 𝔭 generated by f(Z_{1}, Z_{2}) := Z_{1}^{2} − Z_{2} ∈ K[Z_{1}, Z_{2}]. While the variables are in Noether position, any specialization of Z_{1} returns a single point, notwithstanding deg(f) = 2. On the otherside, this does not happen for a really generic frame of coordinates: if we perform a generic change of coordinates obtaining

f(Y_{1}, Y_{2}) := d_{11}^{2}Y_{1}^{2} + d_{11}d_{12}Y_{1}Y_{2} + d_{12}^{2}Y_{2}^{2} − d_{21}Y_{1} + d_{22}Y_{2}

each evaluation of Y_{1} returns two points.

More in general we need to avoid a degeneration in which

deg(f(p_{1}, …, p_{δ}, Z_{δ+1}, …, Z_{r}) < deg(f);

this cannot occur if {Z_{0}, Z_{1}, …, Z_{r}} is in Noether position for the homogeneous ideal ^{h}L_{r}; (2) gcd and resultant computation of polynomials have good complexity if

performed over L[T] where L is a field, but this implies the ability of performing zero-testing and inverting, which is out of the present model. The computation, e.g., of the resultant

A(Z_{δ}) := Res(Q, f(Z_{δ}, V_{δ+1}, …, V_{r})

which satisfies deg(A) ≤ η := deg(L^{e})) deg(f) costs

M(η) := O(η log^{2}(η) log log(η))

arithmetical operations in K(Z_{δ}) if it is performed in this field where we don’t have a good complexity model for zero-testing and inverting; the result of this computation gives a polynomial A(Z_{δ}) ∈ K[Z_{δ}] of degree η. Let us now fix a value p ∈ K and, remarking that K(Z_{δ}) ⊂ K[[Z_{δ}]], consider the ring

R := K[[Z_{δ}]]/(Z_{δ} − p)^{η+1} ≅ Span_{K}{1, …, Z_{δ}^{η}}

where we can perform both

• zero-testing: an element g = ∑_{i=0}^{η} c_{i}Z_{δ}^{t} ∈ R is zero iff c_{i} = 0 for each i iff g(p) = 0,

• inverting : the inverse in R of the invertible polynomial g, g(p) ≠ 0, is the polynomial s(Z_{δ}), deg(s) ≤ η which satisfies

s(Z_{δ})g(Z_{δ}) + t(Z_{δ})(Z_{δ} − p)^{η+1} = gcd(g(Z_{δ}), (Z_{δ} − p)^{η+1} = 1

for a suitable t(Z_{δ}), deg(t) < deg(g). Thus any element

g(Z_{δ}) := d(Z_{δ})/r(Z_{δ}) ∈ K(Z_{δ}), d(Z_{δ}), r(Z_{δ}) ∈ K[Z_{δ}]

can be canonically represented by an element ġ ∈ Span_{K}{1, …, Z_{δ}^{η}} such that ġ(Z_{δ})r(Z_{δ}) ≡ d(Z_{δ}) mod (Z_{δ} − p)^{η+1}. Thus the same algorithm which, if performed on K(Z_{δ}), resturns A(Z_{δ}) with complexity M(η) can be performed also on R with the same complexity M(η) returning some polynomial B(Z_{δ}) := ∑_{i=0}^{η} c_{i}Z_{δ}^{t} ∈ K[Z_{δ}] of degree bounded by η. Can we assume that such polynomial B(Z_{δ}) is the true resultant A(Z_{δ}), i.e. that B(Z_{δ}) = Ȧ(Z_{δ}) = A(Z_{δ})? The answer is obvious: the solution is correct iff in each step of the computation, the algorithm in R gives the same answer as the algorithm in K(Z_{δ}), id est

• when zero-testing g(Z_{δ}) ∈ K(Z_{δ}), g = 0 ⟺ ġ(p) = 0,

• when computing the inverse h(Z_{δ}) := g^{−1}(Z_{δ}) of g(Z_{δ}) ∈ K(Z_{δ}), the polynomial s(Z_{δ}) such that s(Z_{δ})ġ(Z_{δ}) ≡ 1 mod (Z_{δ} − p)^{η+1} satisfies s = ḣ.

This behavieur depends on the choice of p ∈ K; there are some values p in which some wrong answer is returned failing this approach; but almost all choices are ”lucky” thus allowing to produce the correct answer B(Z_{δ}) = Ȧ(Z_{δ}) = A(Z_{δ}) with the good M(η) complexity while computing in the ring R instead than in the field K(Z_{δ}).

(3) In a similar way a better complexity is obtained if the computation (see Remark 44.6.5) of the resultant A(Z) ∈ K[Z] of the polynomials q̂_{t}(Z, T) and f(λ_{δ}^{−1}(Z−T), V̂_{δ+1}(Z, T), …, V̂_{r}(Z, T)) in K[Z][T] can be performed with the ring arithmetics of K[Z] instead of the field arithmetics of K(Z). Such ability depends on a lucky choice of the Liouville point λ_{δ} and of the primitive U.

44.9 Complexity consideration

We record here the following^{5}

Fact 44.9.1. Let R be an integral domain and K be a field. Denoting

M(n) := O(n log^{2}(n) log log(n)),

the following holds

(1) The bit-complexity of the arithmetic operations (addition, multiplicatation, quotient, remainder and gcd) of integers of bit-size^{6} n cost M(n).

(2) Multiplication and division of polynomials in R[T] whose degree is bounded by n cost O(n log(n) log log(n)) arithmetical operations in R.

(3) gcd and resultant computation of polynomials in K[T] whose degree is bounded by n cost M(n) arithmetical operations in K.

(4) Multiplication of two n-square matrices in R costs O(n^{ω}) arithmetical operations in R, with ω < 2.39.

(5) Inversion of an n-square matrix in K costs O(n^{ω}) arithmetical operations in K.

(6) If D = k[T]/q(T) where k is a field and q a squarefree monic polynomial, inversion of an n-square matrix in D costs O(n^{Ω}) arithmetical operations (addition, multiplication, determinant, adjoint matrix) in D where Ω < 4;

5 Cf.

• Aho A.V., Hopcroft J.E., Ullman J.D., The design and analysis of computer algorithms, Addison–Wesley (1974)

• Bini D., Pan V, Polynomial and matrix computations Birkhäuser (1994)

• Bürgisser P., Clausen M., Shorolahi M.A., Algebraic Complexity Theory, Springer (1997)

6 i.e. integers m such that log(m) ≤ n.

(7) performing a linear substitution into a polynomial p ∈ K[X] of degree d

costs M(d) arithmetical operations in K.  �

Definition 44.9.2. A polynomial f ∈ k[X_{1}, …, X_{n}] is said to be given by a straight-line program of size L if there is a sequence {Q_{1}, …, Q_{L}} ⊂ k[X_{1}, …, X_{n}] where f ∈ {Q_{1}, …, Q_{L}} and, for each i, 1 ≤ i ≤ L either

• Q_{i} ∈ {X_{1}, …, X_{n}},

• Q_{i} ∈ k, or

• there are j_{1}, j_{2} < i such that, either – Q_{i} = Q_{j_{1}} + Q_{j_{2}},

– Q_{i} = Q_{j_{1}} − Q_{j_{2}},

– Q_{i} = Q_{j_{1}} · Q_{j_{2}},  �

Theorem 44.9.3. Let

f_{1}, …, f_{r}, g ∈ K[Z_{1}, …, Z_{r}]

be polynomials of degree bounded by D and given by a straight-line program of size at most L; with the same notation and assumptions as in Problem 44.1.5, a geometric resolution of Z_{r} can be computed with

O(r(rL + r^{Ω})M^{2}(DS))

arithmetic operations in K where

S := max(deg(Z_{ρ}), 1 ≤ ρ < r) ≤ D^{r−1}

by means of a probabilistic algorithm.

Its probability of returning correct results relies on choices of elements of K; choices which give a non correct result are contained into a closed Zariski set.

Proof (sketch). Let us remark that

• The computation of Proposition 44.3.1 costs O((rL + r^{Ω})M(S)a(2)) where a(j) is the cost of arithmetical operations in R/I^{j} :

– the evaluation of f and J has complexity O(rL),

– the inversion of J costs O(r^{Ω}),

– the updating of Q and V costs O(r^{2}),

all these costs being evaluated in terms of arithmetical operations in R/I^{2}[T] each such operation costing M(S)a(2)).

• The computation of Corollary 44.3.2 costs O((rL+r^{Ω})M(S) ∑_{j=0}^{log_{2}(ϵ)} a(2^{j})).

• The \underline{Lifting Step} costs O((rL + r^{Ω})M^{2}(S)) since a(j) = M(j) so that

∑_{j=0}^{log_{2}(S+1)} M(2^{j}) ≤ M(S) ∑_{j=0}^{log_{2}(S+1)} 1/2^{j} ∈ O(M(S)

• The \underline{Intersection Step} costs O(r(L + r^{2})M(S)M(dS)) :

– if p ∈ K[Z_{δ}, U] is stored in a two dimensional array of size O(S^{2}), S := deg(p), the computation of p̂ costs O(SM(S)) arithmetical operations in K: it is sufficient to compute p̂_{i} for each homogeneos component p_{i} of p of degree i; in this setting, since also p̂_{i} is homogeneous of degree i, it is sufficient to compute p̂_{i}(Z_{δ}, 1) = p_{i}(λ_{δ}^{−1}(Z_{δ} − 1), 1) i.e. to perform a linear tranformation over each univariate polynomial p_{i}(Z_{δ}, 1) ∈ K[Z_{δ}]; thus the cost is in O (∑_{i=0}^{S} M(i)) ⊂ O(SM(S))

– the computation of Res(Q, f(Z_{δ}, V_{δ+1}, …, V_{r}) costs

O((L + r^{2})M(s)M(ds)) ≤ O((L + r^{2})M(S)M(DS))

arithmetical operations in K, where L is the size of the f, d := deg(f) ≤ D, s := deg(Q) ≤ S : the computation is in fact performed in k[[Z_{δ}]]/((Z_{δ} − p)^{ds+1});

– the computation of A_{t}(Z) costs O((r − D)(L + r^{2})M(S)M(DS)) arithmetical operations in K: apply the result above to

K[T_{δ}, T_{δ+1}, …, T_{r}]/(T_{δ}, T_{δ+1}, …, T_{r})^{2};

– the computation of the geometric resolution of L_{t} costs O(r^{2}M(S)M(DS)) arithmetical operations in K: ∘ the arithmetics in L := (K[Z_{δ}]/(Z_{δ} − p)^{ds+1}) costs

O(M(ds)) ≤ O(M(DS)),

∘ the computation of the polynomials W_{δ+i} requires O(rM(S)) arith-

metical operations in L ∘ the computation of the polynomials V_{t,d+i} requires O(r) arithmeti-

cal operations in K[T_{δ}, T_{δ+1}, …, T_{r}]/(q_{t}) + (T_{δ}, T_{δ+1}, …, T_{n})^{2} which means O(r^{2}M(S)) arithmetical operations in L and O(r^{2}M(S)M(DS)) in K;

∘ removing multiplicity costs O(rM(s)), s := deg(A_{t}) ≤ S.

• The \underline{Cleaning Step} costs O((L + r^{2})M(S)).  �

Remark 44.9.4. Therefore ’generic’ choices give the complete answer.

Moreover the result can be checked by evaluating the input polynomial; if they satisfy the required equations, at most the algorithm failed to recover all roots.

In the special case in which g = 1, i.e. the case in which we want to compute the roots Z(f_{1}, …, f_{r}), since Bezout’s theorem informs that the number of solutions is ∏_{i=1}^{r} deg(f_{i}), it is therefore possible to check whether

the algorithm recovered all solutions.  �

It is worthwhile to compare the complexity of this algorithm, with a Gröbner basis approach; let us therefore assume that each polynomial is given by a dense representation:

Corollary 44.9.5. Let

f_{1}, …, f_{r}, ∈ K[Z_{1}, …, Z_{r}]

be polynomials of degree bounded by D ≥ n and let g := 1; with the same notation and assumptions as in Problem 44.1.5, a geometric resolution of Z_{r} can be computed with O(D^{3(r+O(1)})) arithmetic operations in K by means of the probabilistic algorithm of Theorem 44.9.3.

Proof. By Bezout’s theorem DS ≤ D^{r} so that M(DS)) is in D^{r+O(1)} and

L ≤ r\binom{D+r}{r} which is in D^{r+O(1)} too.  �

Remark 44.9.6. We recall that the degree bound of a 0-dimensional Gröbner basis is, with the present notation, O(D^{r}) (cf. Sections 38.3 and 38.4) so that the dense representation of each polynomial costs γ := O(D^{r^{2}}) and a rougth evaluation of the cost of Gröbner basis computation returns O(D^{4r^{2}}) (cf. the

final comments of Chapter 22).  �

250 44. Kronecker IV

45. Duval II

This conclusive chapter of the Part on ‘algebraic solving’ can be devoted to nothing but the application to multivariate systems of the Kronecker’s Philosophy expounded in the first volume that ’solving’ does not mean producing programs which compute the roots of polynomial equation systems; it means producing programs which compute with their roots.

In the multivariate case, given a finitely generated (zero-dimensional) ideal J ⊂ K[Z_{1}, …, Z_{r}], such philosophy will be put in effect if we were able to consider each its root (a_{1}, …, a_{r}) ∈ Z(J) given by means of a proper suitable representation of the ideal itself.

The considerations performed on the univariate case in the first Part (Chapters 8 and 11; in particular Sections 8.2, 8.3 and 11.4) and Gröbner’s reinterpretation of both Kronecker’s Theory and of the Primitive Element Theorem (Section 8.4) in terms of Allgemeine Basissätze (Section 34.2), which explicitly link root representation with lex Gröbner bases and triangular sets, give the leitmotif of this chapter (see also Section 34.5): I will consider the roots (a_{1}, …, a_{r}) ∈ Z(J) given if

• the ideal J ⊂ K[Z_{1}, …, Z_{r}] is represented by means of one of the Gröbnerrelated technques discussed in the Second Volume, so that

K[Z_{1}, …, Z_{r}]/J → ⨁_{(a_{1},...,a_{r})∈Z(J)} K[a_{1}, …, a_{r}],

• and procedures on K[Z_{1}, …, Z_{r}]/J are given which allow to perform (via Duval splitting) the four operations and zero-testing on each K[a_{1}, …, a_{r}].

This approach will be illustrated by the following instances:

(1) the Kronecker–Duval Model, where I have just to quote the approach endorsed by the Project PoSSo^{1};

(2) the representation of \sqrt{J} by means of an Allgemeine Basis (Definition 34.2.2, Equation (42.1));

(3) the representation of J by means of Kronecker’s parametrizations (Section 41.9, Chapter 44) and Rational Universal Representations (Proposition 42.9.3, Definition 42.9.16;

1 Polynomial System Solving, ESPRIT-BRA 6846

(4) the respresentation of J by means of a Gröbner representation (Definition 29.3.3);

(5) the respresentation of J by means of the linear representation w.r.t. the term ordering ≺. (Definition 29.3.3);

45.1 Kronecker–Duval Model

In relation with Kronecker–Duval Model, I have just to suggest to reread Chaptres 5, 6, 12, Section 34.5 and Chapter 42 under the light put by the following quotation^{2}:

The standard method for computing with algebraic numbers consists in working in a tower of fields, each field being defined by the minimal polynomial of an algebraic number, defined over the preceding field. The computation in such a field needs addition, multiplication and extended gcd of univariate polynomials, as well as Euclidean division, and, recursively, similar operations in the smaller fields. [...] [Duval’s] dynamic evaluation may be viewed as a lazy factorization: in representation of algebraic numbers, the reducibility of a polynomial may only posing a problem when testing equalities or when inverting elements. When such an operation is needed, a gcd computation (already needed for inverting) allows to detect if there is a reducibility problem and, in this case, to get a partial factorization of the reducible polynomial. It follows that, for dynamic evaluation, an algebraic number is represented as a root of a (possibly reducible) square free polynomial. When a factorization occurs from a non trivial gcd, the computation splits in two cases, depending on which factor has the algebraic number as a root. Sometimes, one of the cases is irrelevant, but frequently both cases are of interest, and one needs to carry on two independent computations. Thus the domain of computation is a reduced artinian ring which is implemented as a family of towers. The evaluation is dynamic in the sense that the towers change during the computation. [...] Towers are equivalent with special algebraic systems, the triangular systems where each polynomial introduces a new variable. More generally, any finite set of algebraic numbers may be represented as a solution of a zero dimensional algebraic system. POSSO Report R2

2 Whose author is Daniel Lazard

45.2 Allgemeine Representation

Let us consider

• K an infinite, perfect field, where, if p := char(K) ≠ 0, it is possible to extact pth roots;

• K its algebraic closure;

• Q = K[Z_{1}, …, Z_{r}],

• W := {Z_{1}^{a_{1}} ⋯ Z_{r}^{a_{r}} : (a_{1}, …, a_{r}) ∈ ℕ^{r}};

• J ⊂ Q a zero-dimensional ideal;

• J = ⋂_{i=1}^{r} 𝔮_{i} its irredundant primary representation in Q;

• for each i, 1 ≤ i ≤ r – 𝔪_{i} = \sqrt{q_{i}}, the associated maximal prime,

– K_{i} := Q/𝔪_{i}, K ⊂ K_{i} ⊂ K,

– Q_{i} := K_{i}[Z_{1}, …, Z_{r}],

– the irredundant primary representations 𝔮_{i} = ∩_{j=1}^{r_{i}}q_{ij} and 𝔪_{i} = ∩_{j=1}^{r_{i}}m_{ij} in Q_{i},

– the roots b_{ij} := (b_{1}^{(ij)}, …, b_{r}^{(ij)}) ∈ K_{i}^{r} ⊂ K^{r}, 1 ≤ j ≤ r_{i},

– d_{ij} := mult(b_{ij}, J) = deg(q_{ij}) for each j, 1 ≤ j ≤ r_{i}, which satisfy:

(1) m_{ij} = (Z_{1} − b_{1}^{(ij)}, …, Z_{r} − b_{r}^{(ij)}),

(2) the b_{ij}s, 1 ≤ j ≤ r_{i}, are K-conjugate for each i,

(3) up to a renumeration, \sqrt{q_{ij}} = m_{ij},

(4) 𝔪_{i} = m_{ij} ∩ Q,

(5) 𝔮_{i} = q_{ij} ∩ Q,

(6) for each j, l, 1 ≤ j, l ≤ r_{i}, d_{ij} = d_{il} =: d_{i},

(7) r_{i} = deg(𝔪_{i}) = [K_{i} : K],

(8) deg(𝔮_{i}) = d_{i}r_{i},

(9) J = ∩_{i=1}^{r} ∩_{j=1}^{r_{i}} q_{ij}, \sqrt{J} = ∩_{i=1}^{r} ∩_{j=1}^{r_{i}} m_{ij} are the irredundant primary representations in K[Z_{1}, …, Z_{r}],

(10) Z(J) = {b_{ij} : 1 ≤ i ≤ r, 1 ≤ j ≤ r_{j}};

• Y := Z_{1} + ∑_{i=2}^{r} c_{i}Z_{i} an allgemeine coordinate (Definition 34.4.7) for J;

• J^{+} = J + (Y − Z_{1} − ∑_{l=2}^{r} c_{l}Z_{l}) ⊂ K[Y, Z_{1}, …, Z_{r}],

• g_{0} the monic primitive generator of J^{+} ∩ K[Y],

• for each i – 𝔪_{i}^{+} = 𝔪_{i} + (Y − Z_{1} − ∑_{l=2}^{r} c_{l}Z_{l}),

– 𝔮_{i}^{+} = 𝔮_{i} + (Y − Z_{1} − ∑_{l=2}^{r} c_{l}Z_{l}),

– h_{i} ∈ K[Z_{1}] the monic polynomial such that (h_{i}) = 𝔪_{i}^{+} ∩ K[Z_{1}],

– for each j, 1 ≤ j ≤ r_{i} ∘ m_{ij}^{+} = m_{ij} + (Y − Z_{1} − ∑_{l=2}^{r} c_{l}Z_{l}), ∘ q_{ij}^{+} = q_{ij} + (Y − Z_{1} − ∑_{li=2}^{r} c_{l}Z_{l}), ∘ β_{ij} = b_{1}^{(ij)} + ∑_{l=2}^{r} c_{l}b_{l}^{(ij)},

which satisfy:

(11) h_{i} = ∏_{j=1}^{r_{j}} (Y − β_{ij}) for each i, (12) g_{0}(Y) = ∏_{i=1}^{r} h_{i}^{d_{i}} = ∏_{i=1}^{r} ∏_{j=1}^{r_{i}} (Y − β_{ij})^{d_{i}} ,

(13) R := deg(g_{0}) = ∑_{i=1}^{r} r_{i}d_{i} = deg(J),

(14) f_{0} := SQFR(g_{0}) = ∏_{i=1}^{r} h_{i} = ∏_{i=1}^{r} ∏_{j=1}^{r_{i}} (Y − β_{ij}).

As a consequence we have the isomorphisms (defined by canonical projection and chinese remaindering):

Q/J  ≅  Q[Y]/J^{+}  ≅  K[Y]/(g_{0})
↕  ↕  ↕
⨁_{i=1}^{r} Q/𝔮_{i}  ≅  ⨁_{i=1}^{r} Q[Y]/𝔮_{i}^{+}  ≅  ⨁_{i=1}^{r} K[Y]/(h_{i}^{d_{i}})
↕  ↕  ↕
⨁_{i=1}^{r} ⨁_{j=1}^{r_{j}} Q_{i}/q_{ij} ≅ ⨁_{i=1}^{r} ⨁_{j=1}^{r_{j}} Q_{i}[Y]/q_{ij}^{+} ≅ ⨁_{i=1}^{r} K_{i}[Y]/ (Y − β_{ij})^{d_{i}}

and

Q/\sqrt{J} ≅  Q[Y]/\sqrt{J^{+}}  ≅  K[Y]/(f_{0})
↕  ↕  ↕
⨁_{i=1}^{r} Q/𝔪_{i} ≅  ⨁_{i=1}^{r} Q[Y]/𝔪_{i}^{+}  ≅  ⨁_{i=1}^{r} K[Y]/(h_{i})
↕  ↕  ↕
⨁_{i=1}^{r} ⨁_{j=1}^{r_{j}} Q_{i}/m_{ij} ≅ ⨁_{i=1}^{r} ⨁_{j=1}^{r_{j}} Q_{i}[Y]/m_{ij}^{+} ≅ ⨁_{i=1}^{r} K_{i}[Y]/ (Y − β_{ij})

Let us now assume that J is radical, so that g_{0} = f_{0} and the reduced Gröbner basis w.r.t. the lex ordering induced by Y < Z_{1} < … < Z_{r} is the allgemaine basis (see Theorem 34.2.1)

(g_{0}(Y), Z_{1} − g_{1}(Y), …, Z_{r} − g_{r}(Y))

of J^{+} and let us show how to apply it in order to perform arithmetical manipulation over each root b_{ij}:

\underline{canonical representation}: all arithmetical expressions

p(b_{ij}) = p(b_{1}^{(ij)}, …, b_{r}^{(ij)}), p ∈ Q

of each root b_{ij} ∈ Z(J) have a canonical representation

p(b_{ij}) = p̂(β_{ij})

where p̂(Y) := Rem (p(g_{1}(Y), …, g_{r}(Y)), g_{0}(Y)) ∈ K[Y].

Remark 45.2.1. For each q(Y) ∈ K[Y], deg(q) < deg(g_{0}), the polynomial p = q̌ ∈ Q defined by

q̌(Z_{1}, …, Z_{r}) := q(Z_{1} + ∑_{i=1}^{r} c_{i}Z_{i})

satisfies the relations

ˆq̌ = p̂ = q and q̌(b_{ij}) = q(β_{ij}).

� \underline{vector space arithmetics}: given two arithmetical expressions p_{1}, p_{2} ∈ Q of

the roots b_{ij} ∈ Z(J) and values c_{1}, c_{2} ∈ K, the arithmetitical expression p(b_{ij}) := c_{1}p_{1}(b_{ij}) + c_{2}p_{2}(b_{ij}) has the canonical representation p(b_{ij}) = p̂(β_{ij}) where

p̂(Y) := c_{1}p̂_{1}(Y) + c_{2}p̂_{2}(Y)
= c_{1}Rem (p_{1}(g_{1}(Y), …, g_{r}(Y)), g_{0}(Y))
+ c_{2}Rem (p_{2}(g_{1}(Y), …, g_{r}(Y)), g_{0}(Y)) ;

\underline{multiplication}: with the same notation the arithmetitical expression

p(b_{ij}) := p_{1}(b_{ij})p_{2}(b_{ij})

has the canonical representation p(b_{ij}) = p̂(β_{ij}) where

p̂(Y) := Rem ((p̂_{1}(Y)p̂_{2}(Y), g_{0}(Y)) ;

\underline{zero testing}: given an arithmetical expression p ∈ Q we have

p(b_{ij}) = 0 ⟺ p̂(Y) := Rem (p(g_{1}(Y), …, g_{r}(Y)), g_{0}(Y)) = 0;

\underline{inverse and division}: given an arithmetical expression p ∈ Q, our aim is to produce

• a factorization g_{0} = g_{0}^{(0)}g_{0}^{(1)},

• a polynomial q(Y) ∈ K[Y], deg(q) < deg(g_{0}^{(1)}) such that ∘ if g_{0}^{(0)} = g_{0}, 1 = g_{0}^{(1)} then, for each b_{ij} ∈ Z(J), p(b_{ij}) = p̂(β_{ij}) = 0; ∘ if g_{0}^{(1)} = g_{0}, 1 = g_{0}^{(0)} then, for each b_{ij} ∈ Z(J),

p(b_{ij}) = p̂(β_{ij}) ≠ 0 and p^{−1}(b_{ij}) = q̌(b_{ij}) = q(β_{ij});

∘ otherwise, for each b_{ij} ∈ Z(J), we have – p(b_{ij}) = p̂(β_{ij}) = 0 ⟺ g_{0}^{(0)}(β_{ij}) = 0,

– p(b_{ij}) = p̂(β_{ij}) ≠ 0 ⟺ g_{0}^{(1)}(β_{ij}) = 0, in which case

p^{−1}(b_{ij}) = q̌(b_{ij}) = q(β_{ij}),

so that, denoting

• I_{0} := {i : 1 ≤ i ≤ r, h_{i} | g_{0}^{(0)}} = {i : h_{i} | p̂} ⊂ {1, ⋯, r},

• I_{1} := {i : 1 ≤ i ≤ r, i ∉ I_{0}} = {i : h_{i} | g_{0}^{(1)}} = {i : h_{i} ∤ p̂} ⊂ {1, ⋯, r},

• J_{ι} := ∩_{i∈I_{ι}}𝔮_{i}, ι ∈ {0, 1},

• Z_{ι} := {b_{ij} : i ∈ I_{ι}}, ι ∈ {0, 1},

• g_{j}^{(ι)} := Rem(g_{j}, g_{0}^{(ι)}) ∈ K[Y], ι ∈ {0, 1}, 1 ≤ j ≤ r, one has

(a) g_{0}^{(ι)} = ∏_{i∈I_{ι}} h_{i} = ∏_{i∈I_{ι}} ∏_{j=1}^{r_{i}} (Y − β_{ij}), ι ∈ {0, 1}; (b) for ι ∈ {0, 1}, (g_{0}^{(ι)}(Y), Z_{1} − g_{1}^{(ι)}(Y), …, Z_{r} − g_{r}^{(ι)}(Y)) is the All-

gemaine basis of J_{ι}^{+} id est its reduced Gröbner basis w.r.t. the lex ordering induced by Y < Z_{1} < … < Z_{r};

(c) Z_{ι} = Z(J_{ι}), ι ∈ {0, 1};

(d) Z_{0} = {b_{ij} ∈ Z(I) : p(b_{ij}) = 0};

(e) Z_{1} = {b_{ij} ∈ Z(J) : p(b_{ij}) ≠ 0};

(f) J = J_{0} ∩ J_{1},

(g) the following isomorphisms hold

Q/J  ≅  Q[Y]/J^{+}  ≅  K[Y]/(g_{0})
↕  ↕  ↕
Q/J_{0} ⊕ Q/J_{1} ≅ Q[Y]/J_{0}^{+} ⊕ Q[Y]/J_{1}^{+} ≅ K[Y]/(g_{0}^{(0)}) ⊕ K[Y]/(g_{0}^{(1)}).

(h) g_{0}^{(ι)} is squarefree, ι ∈ {0, 1};

(i) J_{ι} = \sqrt{J_{ι}}, ι ∈ \{0, 1\}. In order to produce both the required factorization g_{0} := g_{0}^{(0)}g_{0}^{(1)} and polynomial q(Y) ∈ K[Y], deg(q) < deg(g_{0}^{(1)}), having the properties listed above, we simply apply Lazard’s Theorem 11.3.2 and compute:

• g_{0}^{(0)} := gcd(g_{0}, p̂) ∈ K[Y];

• s, t ∈ K[Y] such that sp̂+ tg_{0} = g_{0}^{(0)};

• g_{0}^{(1)} := _{\frac{g_{0}}{g_{0}^{(0)}}} ;

• u, v ∈ K[Y] such that ug_{0}^{(0)} + vg_{0}^{(1)} = 1;

• q := Rem(su, g_{0}^{(1)}). In fact this computation is simply a reformulation of Lazard’s Theorem 11.3.2: denoting p_{1}(Y) := _{\frac{_{p}(Y)}{g_{0}^{(0)}(Y)}} we have:

• if g_{0}^{(0)}(β_{ij}) = 0 then p̂(β_{ij}) = g_{0}^{(0)}(β_{ij})p_{1}(β_{ij}) = 0;

• if g_{0}^{(1)}(β_{ij}) = 0 then

q(β_{ij})p̂(β_{ij}) = s(β_{ij})u(β_{ij})p̂(β_{ij})
= u(β_{ij})s(β_{ij})p̂(β_{ij}) + u(β_{ij})t(β_{ij})g_{0}(β_{ij})
= u(β_{ij})g_{0}^{(0)}(β_{ij})
= u(β_{ij})g_{0}^{(0)}(β_{ij}) + v(β_{ij})g_{0}^{(1)}(β_{ij})
= 1.

Remark that g_{0}^{(0)} = 1 implies g_{0}^{(1)} = g_{0}, v = 0, u = 1 and

Rem(su, g_{0}^{(1)}) = Rem(s, g_{0}^{(1)}).

45.3 Kronecker Parametrization and Rational Universal Representation

While using the same notation as in the previous section, let us now assume to have a Rational Universal Representation

(χ(Y), γ_{0}(Y), γ_{1}(Y), …, γ_{r}(Y))

of J, so that

Z(J) = {b_{ij} : 1 ≤ i ≤ r, 1 ≤ j ≤ r_{j}}
= \{(\frac{γ_{1}(α)}{γ_{0}(α)}, …, \frac{γ_{r}(α)}{γ_{0}(α)}) : α ∈ K, χ(α) = 0\}

and we denote for each i, j, α_{ij} the root of χ(T) for which

b_{ij} = (\frac{γ_{1}(α_{ij})}{γ_{0}(α_{ij})}, …, \frac{γ_{r}(α_{ij})}{γ_{0}(α_{ij})})

and ψ_{i} := ∏_{j}(T − α_{ij}) for each i observing that we have

χ(T) = ∏_{ij}(T − α_{ij})^{d_{i}} = ∏_{i} ψ_{i}^{d_{i}}.

We also denote γ_{−1}(Y) ∈ K[Y], the unique polynomial which satisfyies

γ_{0}(Y)γ_{−1}(Y) ≡ 1 mod χ, deg(γ_{−1}) < deg(χ).

If J is radical, then χ is squarefree, γ_{0} = χ′, d_{i} = 1 and the representation is a Kronecker parametrization.

With the present notation, if the given RUR is associated to the allgemeine coordinate Z_{1} + ∑_{i=2}^{r} c_{i}Z_{i}, then we have χ = g_{0}, α_{ij} = β_{ij} for each i, j and ψ_{i} = h_{i} for each i.

Let us now show how to adapt the considerations of the previous section in this setting:

\underline{canonical representation}: all arithmetical expressions

p(b_{ij}) = p(b_{1}^{(ij)}, …, b_{r}^{(ij)}), p ∈ Q

of each root b_{ij} ∈ Z(J) have a canonical representation

p(b_{ij}) = \frac{p̂(α_{ij})}{γ_{0}(α_{ij})}

where

p̂(Y) = Rem (p(γ_{1}(Y), …, γ_{r}(Y)), χ(Y)) ∈ K[Y].

\underline{vector space arithmetics}: given two arithmetical expressions p_{1}, p_{2} ∈ Q of the roots b_{ij} ∈ Z(J) and values c_{1}, c_{2} ∈ K, the arithmetitical expression p(b_{ij}) := c_{1}p_{1}(b_{ij}) + c_{2}p_{2}(b_{ij}) has the canonical representation p(b_{ij}) :=

\frac{_{p}(α_{ij})}{γ_{0}(α_{ij})} where

p̂(Y) := c_{1}p̂_{1}(Y) + c_{2}p̂_{2}(Y)
= c_{1}Rem (p_{1}(γ_{1}(Y), …, γ_{r}(Y)), χ(Y))
+ c_{2}Rem (p_{2}(γ_{1}(Y), …, γ_{r}(Y)), χ(Y)) ;

\underline{multiplication}: the arithmetitical expression p_{1}(b_{ij})p_{2}(b_{ij}) has the canonical representation \frac{q(α_{ij})}{γ_{0}(α_{ij})} where

q(Y) := Rem (p̂_{1}(Y)p̂_{2}(Y)γ_{−1}(Y), χ(Y)) ,

so that

\frac{q(α_{ij})}{γ_{0}(α_{ij})} = \frac{p̂_{1}(α_{ij})p̂_{2}(α_{ij})γ_{−1}(α_{ij})}{γ_{0}(α_{ij})}
= \frac{(p_{1}(b_{ij})γ_{0}(α_{ij})) · (p_{2}(b_{ij})γ_{0}(α_{ij})) · γ_{−1}(α_{ij})}{γ_{0}(α_{ij})}
= p_{1}(b_{ij})p_{2}(b_{ij});

\underline{zero testing}: given an arithmetical expression p ∈ Q we have

p(b_{ij}) = 0 ⟺ p̂(Y) = Rem (p(γ_{1}(Y), …, γ_{r}(Y), χ(Y)) = 0;

\underline{inverse and division}: Even while χ is not necessarily squarefree, Lazard’s Theorem 11.3.2 can be applied essentially in the same way in order to compute the required factorization χ := χ^{(0)}χ^{(1)} and the polynomial q(Y) ∈ K[Y], deg(q) < deg(χ^{(1)}), having the required properties. We begin by remarking that of all the data related to the multiplicity of the primary components of J — namely r, r_{i}, d_{i} — the only available to us is^{3}

R = deg(χ) = ∑_{i=1}^{r} r_{i}d_{i}

but that is all we need, since

Lemma 45.3.1. With the present notation, it holds

gcd(χ, p^{R}) = ∏_{i∈I_{0}} ψ_{i}^{d_{i}}

where I_{0} = {i : 1 ≤ i ≤ r, ψ_{i} | p̂}.

3 cf. (13) of the properties listed in page 254.

Proof. In fact, for each i, ψ_{i} | p ⟺ ψ_{i}^{d_{i}} | p^{R} ⟺ ψ_{i}^{d_{i}} | gcd(χ, p^{R}). �

Corollary 45.3.2. Denoting

• χ^{(0)} := gcd(χ, γ_{−1}^{2}p^{R}) ∈ K[Y];

• s, t ∈ K[Y] such that sγ_{−1}^{2}p^{R} + tχ = χ^{(0)};

• χ^{(1)} := \frac{χ}{χ^{(0)}} ;

• u, v ∈ K[Y] such that uχ^{(0)} + vχ^{(1)} = 1;

• q := Rem(usp^{R−1}, χ^{(1)}) then ∘ if χ^{(0)} = χ, 1 = χ^{(1)} then, for each b_{ij} ∈ Z(J), p(b_{ij}) = 0; ∘ if χ^{(1)} = χ, 1 = χ^{(0)} then, for each b_{ij} ∈ Z(J),

p(b_{ij}) = \frac{p̂(α_{ij})}{γ_{0}(α_{ij})} ≠ 0 and p^{−1}(b_{ij}) = \frac{q(α_{ij})}{γ_{0}(α_{ij})};

∘ otherwise, for each b_{ij} ∈ Z(J), we have – p(b_{ij}) = 0 ⟺ χ^{(0)}(α_{ij}) = 0,

– p(b_{ij}) = \frac{_{p}(β_{ij})}{γ_{0}(α_{ij})} ≠ 0 ⟺ χ^{(1)}(α_{ij}) = 0, in which case

p^{−1}(b_{ij}) = \frac{q(α_{ij})}{γ_{0}(α_{ij})}.

Proof. Since for each b_{ij} ∈ Z(J), γ_{0}(α_{ij}) ≠ 0 and γ_{−1}(α_{ij}) ≠ 0, Lazard’s Theorem 11.3.2 grants that:

• denoting p_{1} := \frac{γ_{−1}^{2}p^{R}}{χ_{0}^{(0)}} we have

χ_{0}^{(0)}(α_{ij}) = 0 ⟹ p^{R}(α_{ij}) = χ_{0}^{(0)}(α_{ij})p_{1}(α_{ij})γ_{0}(α_{ij}) = 0.

• χ^{(1)}(α_{ij}) = 0 implies p^{R}(α_{ij}) ≠ 0 and

\frac{p̂(α_{ij})}{γ_{0}(α_{ij})} · \frac{q(α_{ij})}{γ_{0}(α_{ij})}

= p̂(α_{ij})q(α_{ij})γ_{−1}^{2}(α_{ij})
= p̂(α_{ij}) (u(α_{ij})s(α_{ij})p^{R−1}(α_{ij})) γ_{−1}^{2}(α_{ij})
= u(α_{ij})s(α_{ij})γ_{−1}^{2}(α_{ij})p^{R}(α_{ij}) + u(α_{ij})t(α_{ij})χ(α_{ij})
= u(α_{ij})χ^{(0)}(α_{ij})
= u(α_{ij})χ^{(0)}(α_{ij}) + v(α_{ij})χ^{(1)}(α_{ij})
= 1.

�

Therefore denoting

• I_{0} := {i : 1 ≤ i ≤ r, ψ_{i} | χ^{(0)}} = {i : ψ_{i} | p̂} ⊂ {1, ⋯, r}, • I_{1} := {i : 1 ≤ i ≤ r, i ∉ I_{0}} = {i : ψ_{i} | χ^{(1)}} = {i : ψ_{i} ∤ p̂} ⊂ {1, ⋯, r},

• J_{ι} := ∩_{i∈I_{ι}}𝔮_{i}, ι ∈ {0, 1},

• Z_{ι} := {b_{ij} : i ∈ I_{ι}}, ι ∈ {0, 1},

• γ_{j}^{(ι)} := Rem(γ_{j}, χ^{(ι)}) ∈ K[Y], ι ∈ {0, 1}, 0 ≤ j ≤ r, one has

Corollary 45.3.3. With the present notation, the following holds:

(a) χ^{(ι)} = ∏_{i∈I_{ι}} ψ_{i}^{d_{i}} = ∏_{i∈I_{ι}} ∏_{j=1}^{r_{i}} (Y − α_{ij})^{d_{i}}, ι ∈ {0, 1};

(b) for ι ∈ {0, 1}, (χ^{(ι)}(Y), γ_{0}^{(ι)}(Y), γ_{1}^{(ι)}(Y), …, γ_{r}^{(ι)}(Y)) is the Rational Universal Representation of J_{ι};

(c) Z_{ι} = \{(\frac{γ_{1}^{(ι)}(α)}{γ_{0}^{(ι)}(α)}, …, \frac{γ_{r}^{(ι)}(α)}{γ_{0}^{(ι)}(α)}) : α ∈ K, χ^{(ι)}(α) = 0\}, ι ∈ \{0, 1\};

(d) Z_{ι} = Z(J_{ι}), ι ∈ {0, 1};

(e) Z_{0} = {b_{ij} ∈ Z(I) : p(b_{ij}) = 0};

(f) Z_{1} = {b_{ij} ∈ Z(J) : p(b_{ij}) ≠ 0};

(g) J = J_{0} ∩ J_{1}, If moreover J is radical and the given Rational Universal Representation is a Kronecker Parametrization, setting

ξ_{j}^{(ι)} := Rem(γ_{j} \frac{∂χ^{(ι)}}{∂Y} γ_{−1}, χ^{(ι)}) ∈ K[Y], ι ∈ \{0, 1\}, 1 ≤ j ≤ n,

it holds

(h) χ^{(ι)} is squarefree, ι ∈ {0, 1};

(i) J_{ι} = \sqrt{J_{ι}}, ι ∈ \{0, 1\}.

(j) χ^{(ι)} = ∏_{i∈I_{ι}} ψ_{i} = ∏_{i∈I_{ι}} ∏_{j=1}^{r_{i}} (Y − β_{ij}), ι ∈ {0, 1};

(k) for ι ∈ {0, 1},

(χ^{(ι)}(Y), \frac{∂χ^{(ι)}}{∂Y} Z_{1} − ξ_{1}^{(ι)}(Y), …, \frac{∂χ^{(ι)}}{∂Y} Z_{r} − ξ_{r}^{(ι)}(Y))

is the Kronecker Parametrization of J_{ι}. �

Proof. The only non-trivial statements are (b) which is Remark 42.9.18 and (k) for which it is sufficient to remark that, by construction, for each b_{ij} ∈ Z_{ι}, and each l, 1 ≤ l ≤ n, one has

b_{l}^{(ij)} = \frac{γ_{l}(α_{ij})}{γ_{0}(α_{ij})}
= γ_{l}(α_{ij})γ_{−1}(α_{ij})
= \frac{γ_{j}^{(ι)}(α_{ij})\frac{∂χ^{(ι)}}{∂Y} (α_{ij})γ_{−1}(α_{ij})}{\frac{∂χ^{(ι)}}{∂Y} (α_{ij})}

= \frac{ξ_{l}^{(ι)}(α_{ij})}{\frac{∂χ^{(ι)}}{∂Y} (α_{ij})}.

�

45.4 Gröbner representation

Using the same notation as in Section 45.2, let us begin by assuming that the zero-dimensional ideal J, deg(J) = R, is given by means of a Gröbner representation (Definition 29.3.3)

q = {q_{1}, …, q_{R}}, q_{1} = 1, M = M(q) := {(a_{lj}^{(h)}) ∈ K^{R^{2}}, 1 ≤ h ≤ r}

so that

(1) Q/J ≅ Span_{K}(q),

(2) Z_{h}q_{l} = ∑_{j} a_{lj}^{(h)}q_{j} for each l, j, h, 1 ≤ l, j ≤ R, 1 ≤ h ≤ n, in Q/J,

and by recalling that^{4} for each f ∈ Q its Gröbner description (Definition 29.3.3)

Rep(f, q) := (γ(f, q_{1}, q), …, γ(f, q_{R}, q)) ∈ K^{R}

in terms of this Gröbner representation which satisfies

f − ∑_{j} γ(f, q_{j}, q)q_{j} ∈ J

can be efficiently computed both when f is represented as a linear combination of terms in W or via a recursive Horner representation.

In other terms as we already observed in Historical Remark 29.3.4, we can efficiently compute the structure constants

γ_{ij}^{(l)} := γ(q_{i}q_{j}, q_{l}, q)

which satisfy

(3) q_{i}q_{j} = ∑_{l} γ_{ij}^{(l)}q_{l} for each l, j, h, 1 ≤ i, j, l ≤ R.

As a consequence we can adapt Definition 29.3.3 saying that

Definition 45.4.1. A Gröbner representation of J is the assignement of

(a) a K-linearly independent set q = {q_{1}, …, q_{R}},

(b) the set M = M(q) := {(a_{lj}^{(h)}) ∈ K^{R^{2}}, 1 ≤ h ≤ n} of n square matrices

(c) R^{3} values γ_{ij}^{(l)} ∈ K

which satisfy

(1) Q/J ≅ Span_{K}(q),

4 Compare the discussion in Section 29.3 and in particular Algorithm 29.3.8.

(2) Z_{h}q_{l} ≡ ∑_{j} a_{lj}^{(h)}q_{j}, (mod J), for each l, j, h, 1 ≤ l, j ≤ R, 1 ≤ h ≤ n,

(3) q_{i}q_{j} ≡ ∑_{l} γ_{ij}^{(l)}q_{l}, (mod J) for each l, j, h, 1 ≤ i, j, l ≤ R.

The values γ_{ij}^{(l)} := γ(q_{i}q_{j}, q_{l}, q) are called the structural constants of the K-algebra Q/J = Span_{K}(q)

The linear representation of J w.r.t. the term ordering ≺ is the Gröbner

representation (N_{≺}(J), M, γ_{ij}^{(l)}) where q = (N_{≺}(J)).  �

The application of Gröbner representations as a tool for effectively perform Kronecker’s Philosophy requires the solution of the following

Problem 45.4.2. Given

• a zero-dimensional ideal J′ ⊃ J and

• a K-basis q′ = {q′_{1}, …, q′_{S}} ⊂ Span_{K}(q)

such thay Q/J′ = Span_{K}(q′), compute a Gröbner representation of J′. �

We postpone discussing the solution of this problem to the end of the section, after we will have expound our application.

Since a Gröbner representation of J gives the natural arithmetics of a Kalgebra, in order to apply it for carrying into effect Kronecker’s Philosophy, we just need to focus on inversion and division.

On the basis of the discussion in Section 29.3 and in particular Algorithm 29.3.8 we can wlog assume that each arithmetical expression given via a polynomial p is represented either via a recursive Horner representation or as a linear combination of terms in W; thus it can be easily expressed as p = ∑_{ι} γ(p, q_{ι}, q)q_{ι}.

\underline{canonical representation}: all arithmetical expressions

p(b_{ij}) = p(b_{1}^{(ij)}, …, b_{r}^{(ij)}), p ∈ Q

of each root b_{ij} ∈ Z(J) have the canonical representation

p(b_{ij}) = ∑_{ι} γ(p, q_{ι}, q)q_{ι}(b_{ij});

\underline{vector space arithmetics}: given two such arithmetical expressions p_{1}, p_{2} ∈ Q of the roots b_{ij} ∈ Z(J) and values c_{1}, c_{2} ∈ K, the arithmetitical expression p(b_{ij}) := c_{1}p_{1}(b_{ij})+c_{2}p_{2}(b_{ij}) has the canonical representation

∑_{j}(c_{1}γ(p_{1}, q_{ι}, q) + c_{2}γ(p_{2}, q_{ι}, q))q_{ι}(b_{ij});

\underline{multiplication}: with the same notation the arithmetitical expression

p(b_{ij}) := p_{1}(b_{ij})p_{2}(b_{ij})

has the canonical representation

∑_{λ} (∑_{ι} ∑_{κ} γ(p_{1}, q_{ι}, q)γ_{ικ}^{(λ)}γ(p_{2}, q_{κ}, q)) q_{λ}(b_{ij});

since

∑_{λ} (∑_{ι} ∑_{κ} γ(p_{1}, q_{ι}, q)γ_{ικ}^{(λ)}γ(p_{2}, q_{κ}, q)) q_{λ}(b_{ij})

= ∑_{ι} ∑_{κ} γ(p_{1}, q_{ι}, q)γ(p_{2}, q_{κ}, q) (∑_{λ} γ_{ικ}^{(λ)}q_{λ}(b_{ij}))
= ∑_{ι} ∑_{κ} γ(p_{1}, q_{ι}, q)γ(p_{2}, q_{κ}, q)q_{ι}(b_{ij})q_{κ}(b_{ij})

= (∑_{ι} γ(p_{1}, q_{ι}, q)q_{ι}(b_{ij})) · (∑_{κ} γ(p_{2}, q_{κ}, q)q_{κ}(b_{ij}))
= p_{1}(b_{ij})p_{2}(b_{ij}).

\underline{zero testing}: given an arithmetical expression p ∈ Q we have

p(b_{ij}) = 0 ⟺ γ(p, q_{i}, q) = 0 for each i.

\underline{inverse and division}: Linear algebra, namely Gaussian algorithm, is all we need to deal with division. Let us consider any p ∈ Q given by the canonical representation

p ≡ ∑_{ι=1}^{R} γ(p, q_{ι}, q)q_{ι} mod J

and let us remark that, for each i ≤ r, setting

δ := \{ \begin{matrix}1 & J = \sqrt{J} \\ R = deg(J) & J ≠ \sqrt{J} \end{matrix}

we have p(b_{ij}) = 0 ⟺ p ∈ 𝔪_{i} ⟺ p^{δ} ∈ 𝔮_{i} ⟺ 𝔮_{i} : p^{δ} = 1; p(b_{ij}) ≠ 0 ⟺ p ∉ 𝔪_{i} ⟺ p^{δ} ∉ 𝔮_{i} ⟺ 𝔮_{i} : p^{δ} = 𝔮_{i}; p(b_{ij}) = 0 ⟺ 𝔮_{i} + (p^{δ}) = 𝔮_{i} ⟺ 𝔪_{i} + (p) = 𝔪_{i}; p(b_{ij}) ≠ 0 ⟺ 𝔮_{i} + (p^{δ}) = (1) ⟺ 𝔪_{i} + (p) = (1); therefore, denoting

• A the matrix A := (γ(p^{δ−1}q_{κ}, q_{λ}, q))_{λκ} • I_{0} := {i : 1 ≤ i ≤ r, p^{δ} ∈ 𝔮_{i}} = {i : p ∈ 𝔪_{i}} = {i : p(b_{ij}) = 0} ⊂

{1, ⋯, r},

• I_{1} := {i : 1 ≤ i ≤ r, i ∉ I_{0}} = {i : p^{δ} ∉ 𝔮_{i}} = {i : p ∉ 𝔪_{i}} = {i : p(b_{ij}) ≠ 0} ⊂ {1, ⋯, r},

• J_{ι} := ∩_{i∈I_{ι}}𝔮_{i}, ι ∈ {0, 1},

• Z_{ι} := {b_{ij} : i ∈ I_{ι}}, ι ∈ {0, 1};

• b = {b_{1}, …, b_{R}} another K-basis of Q/J = Span_{K}(b),

• ϕ : Q/J ≅ Span_{K}(q) → Q/J ≅ Span_{K}(b) the morphism defined by ϕ(g) = p^{δ}g for each g ∈ Span_{K}(q),

• C := (c_{ικ}) the matrix representing the morphism ϕ,

• π : Q → Q/J ≅ Span_{K}(q) the canonical projection,

• S := deg(J_{0}), we have

(1) J = J_{0} ∩ J_{1};

(2) J_{0} + J_{1} = (1);

(3) J_{0} = J + (p^{δ}), J_{1} = J : p^{δ};

(4) Z_{ι} = Z(J_{ι}), ι ∈ {0, 1};

(5) Z_{0} = {b_{ij} ∈ Z(I) : p(b_{ij}) = 0};

(6) Z_{1} = {b_{ij} ∈ Z(J) : p(b_{ij}) ≠ 0};

(7) ϕ(q_{κ}) = p^{δ}q_{κ} = ∑_{ι=1}^{R} b_{ι}c_{ικ}, 1 ≤ κ ≤ R;

(8) Im(ϕ) ⊂ Q/J is the principal ideal generated by π(p^{δ});

(9) π^{−1}(Im(ϕ)) = J_{0} = J + (p^{δ});

(10) ker(ϕ) = {π(g) : g ∈ Q, π(gp^{δ}) = 0} ⊂ Q/J

(11) ker(ϕπ) = {g ∈ Q : gp^{δ} ∈ J} = J : p^{δ} = J_{1};

(12) S is the rank of C.

Remark 45.4.3. If J is radical, id est δ = 1, then both J_{0} and J_{1} are radical. In this case, the procedure outlined here, which requires the evaluation of p^{δ−1} and p^{δ−1}q′_{ι} in order to compute both p^{δ} and p·(p^{δ−1}q′_{ι}), is simplyfied

since we need just to evaluate pq′_{ι}.  �

If we now perform Gaussian column reduction on C := (c_{ικ}) performing the same transformation on the identity we obtain two matrices

D := (d_{ιℓ}) and E := (e_{κℓ})

which satisfy

(13) D = CE;

(14) E is invertible;

(15) D is lower triangular, so that ℓ > ι ⟹ d_{ιℓ} = 0;

(16) ℓ > S = deg(J_{0}) ⟹ d_{ιℓ} = 0 for each ι;

(17) q′ := {q′_{1}, …, q′_{S}}, q′_{ℓ} := ∑_{ι=1}^{R} b_{ι}d_{ιℓ}, 1 ≤ ℓ ≤ S, is a linear basis both of Im(ϕ) and π(J_{0}) ⊂ Q/J; (18) for each ℓ, 1 ≤ ℓ ≤ S, χ_{ℓ} := ∑_{κ=1}^{R} q_{κ}e_{κℓ} ∈ Span_{K}(q) satisfies

p^{δ}χ_{ℓ} = ϕ(χ_{ℓ}) = q′_{ℓ};

(19) for each ℓ, 1 ≤ ℓ ≤ S, p^{δ−1}χ_{ℓ} has the representation^{5}

p^{δ−1}χ_{ℓ} = ∑_{λ=1}^{R} q_{λ} (∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{λ}, q)e_{κℓ});

(20) q′′ := {χ_{S+1}, …, χ_{R}}, χ_{ℓ} := ∑_{κ=1}^{R} q_{κ}e_{κℓ}, S + 1 ≤ ℓ ≤ R, is a linear basis of both ker(ϕ) and π(J_{1}) ⊂ Q/J;

(21) {q′_{1}, …, q′_{S}} ∪ {χ_{S+1}, …, χ_{R}} is a K-basis of Q/J,

(22) there is a linear relation (mod J)

1 = ∑_{ℓ=1}^{S} c_{ℓ}q′_{ℓ} + ∑_{ℓ=S+1}^{R} c_{ℓ}χ_{ℓ}

= ∑_{ℓ=1}^{S} c_{ℓ}p^{δ}χ_{ℓ} + ∑_{ℓ=S+1}^{R} c_{ℓ}χ_{ℓ}

= p ∑_{ℓ=1}^{S} c_{ℓ} (∑_{λ=1}^{R} q_{λ} (∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{λ}, q)e_{κℓ})) + ∑_{ℓ=S+1}^{R} c_{ℓ}χ_{ℓ}

= p (∑_{λ=1}^{R} q_{λ} (∑_{ℓ=1}^{S} ∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{λ}, q)e_{κℓ}c_{ℓ})) + ∑_{ℓ=S+1}^{R} c_{ℓ}χ_{ℓ};

(23) denoting

η_{λ} := ∑_{ℓ=1}^{S} ∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{λ}, q)e_{κℓ}c_{κ}, 1 ≤ λ ≤ R,

and setting

5 We have

p^{δ−1}χ_{ℓ} = ∑_{κ=1}^{R} p^{δ−1}q_{κ}e_{κℓ}

= ∑_{κ=1}^{R} ∑_{λ=1}^{R} q_{λ}γ(p^{δ−1}q_{κ}, q_{λ}, q)e_{κℓ}

= ∑_{λ=1}^{R} q_{λ} (∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{λ}, q)e_{κℓ}) .

q := ∑_{λ=1}^{R} q_{λ}η_{λ}

we have 1 ≡ qp mod J_{1};

(24) for each κ, S < κ ≤ R, there is a value^{6} ρ(κ) for which e_{ρ(κ)κ} = 1 and e_{ρ(κ)ℓ} = 0 for each ℓ ≠ κ.

As a consequence

Corollary 45.4.4. With the present notation it holds ∘ if S = 0 then C is the null-matrix, J_{1} = (1), J_{0} = J and p(b_{ij}) = 0; ∘ if S = R then C is invertible, J_{0} = (1), J_{1} = J,

p(b_{ij}) ≠ 0 and p^{−1}(b_{ij}) = q(b_{ij});

∘ otherwise, for each b_{ij} ∈ Z(J), we have – p(b_{ij}) = 0 ⟺ b_{ij} ∈ Z(J_{0}),

– p(b_{ij}) ≠ 0 ⟺ b_{ij} ∈ Z(J_{1}), in which case

p^{−1}(b_{ij}) = q(b_{ij}).

�

In conclusion Gaussian reduction is all we need in order to obtain the Duval splitting J = J_{0} ∩ J_{1}; what we have to do is:

• compute the canonical representation of p^{δ−1};

• compute the matrix A representing the canonical representation of p^{δ−1}q_{κ}, 1 ≤ κ ≤ R;

• compute the canonical representation of

p^{δ} = p · p^{δ−1} and p · (p^{δ−1}q_{κ}), 1 ≤ κ ≤ R;

thus obtaining the matrix C;

• perform Gaussian column-reduction on C deducing D and E;

• extract the K-linearly independent bases q′ and q” of, respectively, π(J_{0}) and π(J_{1});

• apply the solution discussed below of Problem 45.4.2 in order to obtain a Gröbner representation of both J_{0} and J_{1};

• compute, for each κ, λ, 1 ≤ κ ≤ S, 1 ≤ λ ≤ R, the values

ϵ_{κ,λ} := ∑_{ι=1}^{R} e_{ικ}γ(p^{δ−1}q_{ι}, q_{λ}, q);

6 Remark that the κ^{th} column of D, which is null, corresponds to a repeated transformation of the ρ(κ)^{th} column of C which is never chosen as a pivot element.

• compute the unique solution (c_{1}, …, c_{R}) of the linear equations

\{ \begin{matrix}1 & = & ∑_{ℓ=1}^{S} c_{ℓ} (∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{1}, q)e_{κℓ}) + ∑_{ℓ=S+1}^{R} c_{ℓ}e_{1ℓ} \\ 0 & = & ∑_{ℓ=1}^{S} c_{ℓ} (∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{2}, q)e_{κℓ}) + ∑_{ℓ=S+1}^{R} c_{ℓ}e_{2ℓ} \\ & & ⋯ \\ 0 & = & ∑_{ℓ=1}^{S} c_{ℓ} (∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{R}, q)e_{κℓ}) + ∑_{ℓ=S+1}^{R} c_{ℓ}e_{Rℓ} \end{matrix}

• compute η_{λ} := ∑_{ℓ=1}^{S} ∑_{κ=1}^{R} γ(p^{δ−1}q_{κ}, q_{λ}, q)e_{κℓ}c_{κ}, 1 ≤ λ ≤ R,

• return q := ∑_{λ=1}^{R} q_{λ}η_{λ}.

Remark 45.4.5. It is clear that this algorithm is essentially a refinement of Traverso’s Algorithm 29.3.8 for computing J_{0} extended and adapted in order

to obtain also J_{1} := π^{−1}(ker(ψ) and q.  �

Example 45.4.6. Let us consider the ideal

J := 𝕀(Z_{1}^{3} − Z_{1}^{2}, Z_{1}Z_{2}, Z_{2}^{2} − Z_{2}) ⊂ K[Z_{1}, Z_{2}] = Q

whose roots are {(0, 0), (1, 0), (0, 1)}, (0, 0) having multiplicity 2 and the primary component (X_{1}^{2}, X_{2}), the Lagrange basis

q_{1} := 1 − Z_{1}^{2} − Z_{2}, q_{2} := Z_{1} − Z_{1}^{2}, q_{3} := Z_{1}^{2}, q_{4} := Z_{2},

and the polynomial p := 1 − Z_{1} + Z_{2} = q_{1} − q_{2} + 2q_{4}.

We thus have

A := ( \begin{matrix}& q_{1} & q_{2} & q_{3} & q_{4} \\ q_{1} & 1 & 0 & 0 & 0 \\ q_{2} & −3 & 1 & 0 & 0 \\ q_{3} & 0 & 0 & 0 & 0 \\ q_{4} & 0 & 0 & 0 & 8 \end{matrix} ) and C := ( \begin{matrix}& q_{1} & q_{2} & q_{3} & q_{4} \\ Z_{1}^{2} & 3 & −1 & 0 & 0 \\ Z_{2} & −1 & 0 & 0 & 16 \\ Z_{1} & −4 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \end{matrix} )

and we deduce

D := ( \begin{matrix}Z_{1}^{2} & 1 & 0 & 0 & 0 \\ Z_{2} & 0 & 1 & 0 & 0 \\ Z_{1} & 0 & 0 & 1 & 0 \\ 1 & −1 & 0 & −1 & 0 \end{matrix} ) E := ( \begin{matrix}q_{1} & −1 & 0 & −1 & 0 \\ q_{2} & −4 & 0 & −3 & 0 \\ q_{3} & 0 & 0 & 0 & 1 \\ q_{4} & −\frac{1}{16} & \frac{1}{16} & −\frac{1}{16} & 0 \end{matrix} )

Thus we have

• J_{0} = J + 𝕀(p^{3}) = J + {q′_{1}, q′_{2}, q′_{3}} = J + {Z_{1}^{2} − 1, Z_{2}, Z_{1} − 1} = 𝕀(Z_{2}, Z_{1} − 1),

• Z(J_{0}) = {(1, 0)} = Z_{0}

• ϕ(χ_{1}) = ϕ(−q_{1} − 4q_{2} − \frac{1}{16}q_{4}) = ϕ(5Z_{1}^{2} + \frac{15}{16}Z_{2} − 4Z_{1} − 1) = q′_{1},

• ϕ(χ_{2}) = ϕ(\frac{1}{16}q_{4}) = ϕ(\frac{1}{16}Z_{2}) = q′_{2},

• ϕ(χ_{3}) = ϕ(−q_{1} − 3q_{2} − \frac{1}{16}q_{4}) = ϕ(4Z_{1}^{2} + \frac{15}{16}Z_{2} − 3Z_{1} − 1) = q′_{3},

• AE = ( \begin{matrix}q_{1} & −1 & 0 & −1 & 0 \\ q_{2} & −1 & 0 & 0 & 0 \\ q_{3} & 0 & 0 & 0 & 0 \\ q_{4} & −\frac{1}{2} & \frac{1}{2} & −\frac{1}{2} & 0 \end{matrix} ) and

p^{3}χ_{1} = −q_{1} − q_{2} − \frac{1}{2}q_{4}, p^{3}χ_{2} = \frac{1}{2}q_{4}, p^{3}χ_{3} = −q_{1} − \frac{1}{2}q_{4};

• J_{1} = J : p^{3} = J + {χ_{4}} = J + {q_{3}} = J + {Z_{1}^{2}} = 𝕀(Z_{1}^{2}, Z_{1}Z_{2}, Z_{2}^{2} − Z_{2}),

• Z(J_{1}) = {(0, 0), (0, 1)} = Z_{1},

• {q′_{1}, q′_{2}, q′_{3}, χ_{4}} = {Z_{1}^{2} − 1, Z_{2}, Z_{1} − 1, Z_{1}^{2}} is a K-basis of Q/J;

• 1 = −q′_{1} + χ_{4};

• (η_{1}, …, η_{R}) = AE(−1, 0, 0, 1)^{T} = (1, 1, 0, \frac{1}{2})

• q := q_{1} + q_{2} + \frac{1}{2}q_{4} = −2Z_{2}^{2} − \frac{1}{2}Z_{2} + Z_{1} + 1 satisfies

1 ≡ qp mod J_{1}.

• ρ_{4} = 3, e_{3ℓ} = 0, ℓ ≠ 4.

Example 45.4.7. Let us consider the radical ideal J ⊂ K[Z_{1}, Z_{2}, Z_{3}] discussed in Example 39.2.3, 40.3.2 and 42.8.8 for which we choose

q := N(J) := {1, Z_{1}, Z_{2}, Z_{3}, Z_{1}^{2}, Z_{1}Z_{2}, Z_{2}^{2}, Z_{1}Z_{3}, Z_{3}^{2}}

and the algebraic expression p = Z_{1}Z_{3} so that we have

Z_{0} = Z(J_{0}) = {b_{i} : i ∈ {1, 2, 4, 9}}, Z_{1} = Z(J_{1}) = {b_{i} : i ∈ {3, 5, 6, 7, 8}}.

We thus have

C := ( \begin{matrix}& 1 & Z_{1} & Z_{2} & Z_{3} & Z_{1}^{2} & Z_{1}Z_{2} & Z_{2}^{2} & Z_{1}Z_{3} & Z_{3}^{2} \\ Z_{3}^{2} & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 4 & 6 \\ Z_{1}Z_{3} & 1 & 3 & −1 & 4 & 7 & −1 & −1 & 12 & 7 \\ Z_{2}^{2} & 0 & 3 & −3 & 15 & 9 & −3 & −3 & 42 & 36 \\ Z_{1}Z_{2} & 0 & 6 & −3 & 30 & 18 & −3 & −3 & 84 & 72 \\ Z_{1}^{2} & 0 & 1 & 2 & 0 & 3 & 2 & 2 & 1 & 0 \\ Z_{3} & 0 & −2 & 2 & −8 & −6 & 2 & 2 & −24 & −18 \\ Z_{2} & 0 & −9 & 9 & −45 & −27 & 9 & 9 & −126 & −108 \\ Z_{1} & 0 & −3 & −3 & −3 & −9 & −3 & −3 & −12 & −6 \\ 1 & 0 & 2 & −2 & 6 & 6 & −2 & −2 & 20 & 12 \end{matrix} ) ,

45.4 Gröbner representation 269

whence we compute D := ( \begin{matrix}Z_{3}^{2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ Z_{1}Z_{3} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ Z_{2}^{2} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ Z_{1}Z_{2} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ Z_{1}^{2} & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ Z_{3} & 1 & 0 & \frac{−2}{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ Z_{2} & 0 & 0 & −3 & 0 & 0 & 0 & 0 & 0 & 0 \\ Z_{1} & 1 & 0 & \frac{−1}{3} & 0 & −2 & 0 & 0 & 0 & 0 \\ 1 & −2 & 0 & \frac{2}{3} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} )

and E := ( \begin{matrix}1 & \frac{−7}{6} & 1 & \frac{−37}{9} & 2 & \frac{−8}{3} & 2 & 0 & 0 & 4 \\ Z_{1} & \frac{5}{6} & 0 & \frac{17}{9} & −1 & \frac{4}{3} & −3 & 0 & 0 & −1 \\ Z_{2} & 0 & 0 & \frac{−2}{3} & \frac{1}{3} & 0 & 0 & −1 & −1 & 0 \\ Z_{3} & \frac{13}{6} & 0 & \frac{10}{9} & \frac{−2}{3} & \frac{2}{3} & 0 & 0 & 0 & −5 \\ Z_{1}^{2} & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ Z_{1}Z_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ Z_{2}^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ Z_{1}Z_{3} & \frac{−5}{6} & 0 & \frac{−5}{9} & \frac{1}{3} & \frac{−1}{3} & 0 & 0 & 0 & 1 \\ Z_{3}^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) .

Moreover

1 = −\frac{1}{2}(Z_{3}^{2} + Z_{3} + Z_{1} − 2) − \frac{1}{2}Z_{1}Z_{3} − \frac{3}{2}(3Z_{2}^{2} − 2Z_{3} − 9Z_{2} − Z_{1} + 2)

−9Z_{1}Z_{2} − \frac{1}{2}(Z_{1}^{2} − 2Z_{1})

+ \frac{1}{2}(Z_{1}^{2} − 3Z_{1} + 2) + 9(Z_{1}Z_{2} − Z_{2})

+\frac{9}{2}(Z_{2}^{2} − Z_{2}) + \frac{1}{2}(Z_{3}^{2} + Z_{1}Z_{3} − 5Z_{3} − Z_{1} + 4)

so that

J_{0} = 𝕀(Z_{3}^{2} + Z_{3} + Z_{1} − 2, Z_{1}Z_{3}, 3Z_{2}^{2} − 2Z_{3} − 9Z_{2} − Z_{1} + 2, Z_{1}Z_{2}, Z_{1}^{2} − 2Z_{1})
J_{1} = 𝕀(Z_{1}^{2} − 3Z_{1} + 2, Z_{1}Z_{2} − Z_{2}, Z_{2}^{2} − Z_{2}, Z_{3}^{2} + Z_{1}Z_{3} − 5Z_{3} − Z_{1} + 4)
q = 24^{−1} (2Z_{3}^{2} + 4Z_{1}Z_{3} + 33Z_{2}^{2} + 66Z_{1}Z_{2} − 18Z_{1}^{2} − 20Z_{3} − 99Z_{2} + 38Z_{1} + 18)

�

Let us now finally discuss a solution of Problem 45.4.2:

• for each l, λ, h compute the values

b_{lλ}^{(h)} := ∑_{i=1}^{R} γ(q′_{l}, q_{i}, q)a_{iλ}^{(h)}

which satisfy, for each l, h, (mod J′)

∑_{λ=1}^{R} b_{lλ}^{(h)}q_{λ} = ∑_{λ=1}^{R} (∑_{i=1}^{R} γ(q′_{l}, q_{i}, q)a_{iλ}^{(h)}) q_{λ} ≡ ∑_{i=1}^{R} γ(q′_{l}, q_{i}, q)Z_{h}q_{i} ≡ Z_{h}q′_{l} ∈ J′;

• by linear algebra, for each l, h compute the unique values a′_{lj}^{(h)}, 1 ≤ j ≤ S satisfying

∑_{j=1}^{S} a′_{lj}^{(h)}q′_{j} = ∑_{λ=1}^{R} b_{lλ}^{(h)}q_{λ} ≡ Z_{h}q′_{l} (mod J′);

• for each i, j, λ compute the values

δ_{ij}^{(λ)} := (∑_{ι} ∑_{κ} γ(q′_{i}, q_{ι}, q)γ_{ικ}^{(λ)}γ(q′_{j}, q_{κ}, q))

which satisfy, for each i, j, (mod J′)

∑_{λ=1}^{R} δ_{ij}^{(λ)}q_{λ} = ∑_{λ=1}^{R} (∑_{ι=1}^{R} ∑_{κ=1}^{R} γ(q′_{i}, q_{ι}, q)γ_{ικ}^{(λ)}γ(q′_{j}, q_{κ}, q)) q_{λ}

≡ ∑_{ι=1}^{R} ∑_{κ=1}^{R} γ(q′_{i}, q_{ι}, q)γ(q′_{j}, q_{κ}, q) (∑_{λ=1}^{R} γ_{ικ}^{(λ)}q_{λ})

≡ ∑_{ι=1}^{R} ∑_{κ=1}^{R} γ(q′_{i}, q_{ι}, q)γ(q′_{j}, q_{κ}, q)q_{ι}q_{κ}
= q′_{i}q′_{j} ∈ J′;

• by linear algebra, for each i, j compute the unique values γ′_{ij}^{(l)}, 1 ≤ l ≤ S satisfying

∑_{l=1}^{S} γ′_{ij}^{(l)}q′_{l} = ∑_{λ=1}^{R} δ_{ij}^{(λ)}q_{λ} ≡ q′_{i}q′_{j} mod J′;

thus the data

(a) q′ = {q′_{1}, …, q′_{S}},

(b) M(q′) := {(a′_{lj}^{(h)})

(c) the structure constants γ′_{ij}^{(l)}

are the required Gröbner representation of J′.

45.5 Linear representation

Let us now specialize the results of the previous section to the case in which

(A) q = N_{≺}(J) ordered so that 1 = q_{1} ≺ q_{2} ≺ ⋯ ≺ q_{R}; (B) at each step of the Gaussian algorithm, as pivot element among all pos-

sible choices we systematycally choose the mostleft available column; (C) b = N_{≺}(J) ordered so that b_{1} ≻ ⋯ ≻ b_{R−1} ≻ b_{R} = 1, so that q_{i} =

b_{R−i+1} for each i.

Remark 45.5.1. Given any Gröbner representation of J and a term ordering ≺, a direct application of Möller’s Algorithm 28.2.7 to the functionals

γ(·, q_{ı}, q) returns N_{≺}(J).  �

Example 45.5.2. Conditions (A-B) are satisfied by both Example 45.4.6 and 45.4.6 which further satisfies also condition (C).

Therfore the reader can easily veryfy our claims on thise examples. �

As a consequence of (B) in the transformation of C into D each column, which is not a pivot element, is modified only by means of columns to its left. Therefore, since in (24) the set J := {ρ(κ) : S < κ ≤ R} denotes the indices corresponding to the columns of C which have not being used as pivot element, the assumption (B) allows to reformulate (24) as:

(24)’ denoting ρ(κ) := max{ℓ : e_{ℓκ} ≠ 0} for each κ, S < κ ≤ R, and J := {ρ(κ) : S < κ ≤ R} we have e_{ℓκ} = 0 for each ℓ ≠ κ).

Thus we trivially have

Corollary 45.5.3. If assumptions (A-B) are satisfied, then

N(J_{1}) = {b_{i}, 1 ≤ i ≤ R, i ∉ J}.

�

In the same mood, as a direct consequence of the ordered imposed on b by condition (C) we also get

Corollary 45.5.4. If assumptions (A-C) are satisfied, then

N(J_{0}) = {b_{S+1}, …, b_{R}}.

�

Index

Admissible sequence

– Lazard, 160

– Lazard (weak), 161

– Ritt, 157

– Ritt (strong), 157

– weak, 154 Allgemeine coordinate, 9, 56 Anti-differential operator, 42 Ascending set, 147

Basic set, 148 Bezoutian

– matrix, 107

– polynomials

– – primary , 106

– – secondary, 108 Bezoutiant, 107, 108 Bezoutic

– Emanant, 107

– matrix, 107

– square, 107

Cauchy modules, 204 Chain, 141, 147 Characteristic set, 142, 148 – Ritt, 157

– Wu, 157 Class, 140, 147 Connected, 124 Constant, 138 Curvilinear primary, 60

Differential

– field, 138

– ideal, 139

– indeterminate, 138

– polynomial, 138 Differentiation, 138 Dixon

– matrix, 115

– polynomial, 115 – resultant, 115

Eliminant, 114

Extraneous factor, 74, 80, 81

Faithful

– representation of a group, 195

Geometric resolution, 227 Gröbner description, 8, 30 Gröbner representation, 8, 30, 261

Incidence number, 198

Inertia, 109 Initial, 143, 148 Irreducible spread, 94 Isobaric, 66

Kronecker

– parametrization, 95, 182, 227 – Resolvent, 91

– substitution, 83

Lagrange

– basis, 30

– Resolvent, 202, 212 Lifting fiber, 230 Lifting point, 230 Lifting system, 229 linear representation, 262 Liouville

– point, 239

– substitution, 87, 92

Macaulay’s matrix, 69, 85

Mark, 197, 198

Matrix

– Bezoutian, 107

– Bezoutic, 107

– Dixon, 115

– Hankel, 103 – Macaulay’s, 69

– Sylvester, 103 Median set, 173

Non-derogatory

– family, 53

– ideal, 53

– matrix, 35

Quasi-component, 161 Quasi-prime ideal, 161

Rank

– of a chain, 141

– of diffienential polynomials, 140 Rational Univariate Representation

(RUR), 189 Reduced, 140, 141, 147

– Initially, 157 Regular, 90 Regular set, 161 Remainder, 145, 149 Resolvent

– Kronecker, 91

– Lagrange, 202, 212 Resultant, 66

– Dixon, 115

– Macaulay’s, 70, 85

Separant, 143 Separating element, 36, 181 Squarefree decomposition, 188 Straight-line program, 247 Structure constants, 262 Substitution

– Kronecker, 83

– Liouville, 87

Tower of simple extensions, 161 Transitive, 195 Triangular set, 157

– fine, 157

– initially reduced, 157

– reduced, 147 Triangularizable ideal, 165

u-resolvent, 92 u-resultant, 88

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