A boring section on notations

Let $\cal {P}$ : = k[X1,…, Xn], $\sf m$ = (X1,…, Xn) the maximal ideal at the origin, $\cal {T}$ : = {X1a1 ... Xnan : (a1,…, an)∈$\mathbb {N}$n}, < the lexicographical ordering on $\cal {T}$ induced by X1 < ... < Xn.

We denote $\sf k$ the algebraic closure of k; for each zero-dimensional ideal $\sf I$⊂$\cal {P}$, $\cal {Z}$($\sf I$) : = {$\sf a$∈$\sf k^{n}_{}$ : f ($\sf a$) = 0,∀f∈$\sf I$}⊂$\sf k^{n}_{}$; for any α = (b1,…, bd)∈$\sf k^{d}_{}$, Φα the projection Φα : $\cal {P}$ $\mapsto$ $\sf k$[Xd+1,…, Xn] defined by

Φα(f )= f (b1,…, bd, Xd+1,…, Xn]∀f∈k[X1,…, Xn].

Each element f∈$\cal {P}$ can be uniquely expressed either as

f = $\displaystyle \sum_{{i=0}}^{{\deg(f)}}$giXni∈k[X1,…, Xn-1][Xn],

gi∈k[X1,…, Xn-1], gdeg(f)≠0, or as a linear combination

f = $\displaystyle \sum_{{t\in{{\cal T}}}}^{}$c(f, t)t = $\displaystyle \sum_{{i=1}}^{s}$c(f, ti)ti,

c(f, ti)≠0, ti∈$\cal {T}$, t1 > ... > ts of terms t∈$\cal {T}$ with coefficients c(f, t) in k; and we will denote Lp$\nolimits$(f ): = gdeg(f) the leading polynomial of f, $\bf T$(f ): = t1 its maximal term, lc$\nolimits$(f ): = c(f, t1) its leading cofficient.

For each set G⊂$\cal {P}$, $\bf T${G} denotes the set {$\bf T$(g) : g∈G}, and $\bf T$(G) the monomial ideal {τ$\bf T$(g) : τ∈$\cal {T}$, g∈G} it generates. For each ideal $\sf I$⊂$\cal {P}$, we denote $\bf G$($\sf I$) the minimal basis of the monomial ideal $\bf T$($\sf I$) = $\bf T${$\sf I$}, $\bf N$($\sf I$) : = $\cal {T}$ $\setminus$ $\bf T$($\sf I$) and

$\displaystyle \bf B$($\displaystyle \sf I$) : = {Xht : 1≤h≤n, t∈$\displaystyle \bf N$($\displaystyle \sf I$)} $\displaystyle \setminus$ $\displaystyle \bf N$($\displaystyle \sf I$)  
  = $\displaystyle \bf T$($\displaystyle \sf I$)∩$\displaystyle \left(\vphantom{\{1\} \cup \{X_h t : 1\leq h \leq n, t\in {\bf N}({\sf I})\}}\right.${1}∪{Xht : 1≤h≤n, t∈$\displaystyle \bf N$($\displaystyle \sf I$)}$\displaystyle \left.\vphantom{\{1\} \cup \{X_h t : 1\leq h \leq n, t\in {\bf N}({\sf I})\}}\right)$  

and we set k[$\bf N$($\sf I$)] : = Span$\nolimits_{k}^{}$($\bf N$($\sf I$)).

For each f∈$\cal {P}$, there is [2,3,4] a unique canonical form

g : = Can$\displaystyle \nolimits$(f,$\displaystyle \sf I$) = $\displaystyle \sum_{{t\in{\bf N}({\sf I})}}^{}$γ(f, t, < )t∈k[$\displaystyle \bf N$($\displaystyle \sf I$)]

such that f - g∈$\sf I$. A Gröbner basis [2,3] of $\sf I$ is any set G⊂$\sf I$ such that $\bf T$(G) = $\bf T${$\sf I$}, i.e. $\bf T${G} generates the monomial ideal $\bf T$($\sf I$) = $\bf T${$\sf I$}; the reduced Gröbner basis [2,3] of $\sf I$ is the set $\cal {G}$($\sf I$) : = {τ - Can$\nolimits$(τ,$\sf I$) : τ∈$\bf G$($\sf I$)}; the border basis [15] of $\sf I$ is the set $\cal {B}$($\sf I$) : = {τ - Can$\nolimits$(τ,$\sf I$) : τ∈$\bf B$($\sf I$)}.

Two sets $\mathbb {L}$ : = {$\ell_{1}^{}$,…,$\ell_{s}^{}$}⊂$\cal {P}$ * : = Hom$\nolimits_{k}^{}$($\cal {P}$, k), and $\bf q$ = {q1,…, qs}⊂$\cal {P}$ are triangular if $\ell_{i}^{}$(qj) = 0, for each i < j.

Denoting, for each k-vector subspace L⊂$\cal {P}$ * ,

$\displaystyle \mathfrak$P(L) : = {g∈$\displaystyle \cal {P}$ : $\displaystyle \ell$(g) = 0, for each $\displaystyle \ell$∈L}

and, for each k-vector subspace P⊂$\cal {P}$,

$\displaystyle \mathfrak$L(P) : = {$\displaystyle \ell$∈$\displaystyle \cal {P}$ * : $\displaystyle \ell$(g) = 0, for each g∈P},

we recall [12,13,16,1,20] that the mutually inverse maps $\mathfrak$L(⋅) and $\mathfrak$P(⋅) give a biunivocal, inclusion reversing, correspondence between the set of the zero-dimensional ideals P⊂$\cal {P}$ and the set of the finite k-dimensional $\cal {P}$-modules L⊂$\cal {P}$ * .

Denoting, for each τ∈$\cal {T}$, M(τ) : $\cal {P}$→k the morphism defined by M(τ) = c(f, τ) for each f = $\sum_{{t\in{{\cal T}}}}^{}$c(f, t)t∈$\cal {T}$ and $\mathbb {M}$ : = {M(τ) : τ∈$\cal {T}$}, then Span$\nolimits_{k}^{}$($\mathbb {M}$)⊂$\cal {P}$ * is the set of the Noetherian equations [12,13,20] of $\cal {P}$. If we set

$\displaystyle \ell$(τ) : = M(τ) + $\displaystyle \sum_{{t\in{\bf T}({\sf I})}}^{}$γ(t, τ, < )M(t)∈Span$\displaystyle \nolimits_{k}^{}$($\displaystyle \mathbb {M}$),

for each $\sf m$-closed ideal $\sf I$⊂$\cal {P}$ and each τ∈$\bf N$($\sf I$), then $\sf I$ can be characterized [12,13,16,20] by its unique Macaulay basis {$\ell$(τ) : τ∈$\bf N$($\sf I$)}, which is a k-basis of the k-subspace {$\ell$∈$\cal {P}$ * : $\ell$(g) = 0,∀g∈$\sf I$}⊂Span$\nolimits_{k}^{}$($\mathbb {M}$)⊂$\cal {P}$ * consisting of all the Noetherian equations of $\sf I$.

Therefore, each zero-dimensional ideal $\sf I$⊂$\cal {P}$ can be considered as given if we know the set $\sf Z$ : = $\cal {Z}$($\sf I$) and, for each $\sf a$∈$\sf Z$, a Macaulay basis of the corresponding primary component of $\sf I$. For each $\sf a$∈$\sf Z$, $\sf a$ : = (a1,…, an), let us therefore denote λ$\scriptstyle \sf a$ : $\cal {P}$ $\mapsto$ $\cal {P}$ the translation λ$\scriptstyle \sf a$(Xi) = Xi + ai, for each i, $\mathfrak$m$\scriptstyle \sf a$ = (X1 - a1,…, Xn - an), $\mathfrak$q$\scriptstyle \sf a$ the $\mathfrak$m$\scriptstyle \sf a$-primary component of $\sf I$, {$\ell$(υ) : υ∈$\bf N_{<}^{}$(λ$\scriptstyle \sf a$($\mathfrak$q$\scriptstyle \sf a$))} is the Macaulay basis of λ$\scriptstyle \sf a$($\mathfrak$q$\scriptstyle \sf a$), $\ell_{{\upsilon{\sf a}}}^{}$ for each υ∈$\bf N_{<}^{}$(λ$\scriptstyle \sf a$($\mathfrak$q$\scriptstyle \sf a$)) the Macaulay equation $\ell_{{\upsilon{\sf a}}}^{}$ : = $\ell$(υ). Setting s : = $\sum_{{{\sf a}\in{\sf Z}}}^{}$deg($\mathfrak$q$\scriptstyle \sf a$) and $\mathbb {L}$ : = {λ1,…, λs} : = {$\ell_{{\upsilon{\sf a}}}^{}$λ$\scriptstyle \sf a$ : υ∈$\bf N_{<}^{}$(λ$\scriptstyle \sf a$($\mathfrak$q$\scriptstyle \sf a$)),$\sf a$∈$\sf Z$}, we know that Span$\nolimits_{k}^{}$($\mathbb {L}$) = $\mathfrak$L($\sf I$) and $\sf I$ = $\mathfrak$P(Span$\nolimits_{k}^{}$($\mathbb {L}$)); moreover [12,13,18,20] we can wlog assume $\mathbb {L}$ to be ordered so that, for each σ, $\mathfrak$P(Span$\nolimits_{k}^{}$({λ1,…, λσ})) is an ideal. We also set

$\displaystyle \sf X$ : = {$\displaystyle \sf x_{1}^{}$,…,$\displaystyle \sf x_{s}^{}$} : = {($\displaystyle \sf a$, υ) : υ∈$\displaystyle \bf N_{<}^{}$($\displaystyle \mathfrak$q$\scriptstyle \sf a$),$\displaystyle \sf a$∈$\displaystyle \sf Z$}

enumerated so that $\sf x_{j}^{}$ = ($\sf a$, υ)$\iff$λj = $\ell_{{\upsilon{\sf a}}}^{}$λ$\scriptstyle \sf a$ and we set, for each j, 1≤j≤s, M(λj) : = M(υ)λ$\scriptstyle \sf a$ where λj = $\ell_{{\upsilon{\sf a}}}^{}$λ$\scriptstyle \sf a$.

Under the assumption that λ = M(λ) for each λ∈$\mathbb {L}$, — this is equivalent as assuming that each λ$\scriptstyle \sf a$($\mathfrak$q$\scriptstyle \sf a$) is a monomial ideal — Cerlienco–Mureddu Algorithm (Alg. [*]) associates to each such sets[*] $\mathbb {L}$ and $\sf X$, an order ideal[*] $\bf N$ : = $\bf N$($\mathbb {L}$) and a bijection Φ : = Φ($\mathbb {L}$) : $\mathbb {L}$ $\mapsto$ $\bf N$, which, as we will proof later, satisfies $\bf N_{<}^{}$($\mathbb {L}$) = $\bf N$($\mathfrak$P($\mathbb {L}$)) for the lexicographical ordering induced by X1 < ... < Xn.

Definition 1.1   The ordered sets $\mathbb {L}$($\sf I$) : = $\mathbb {L}$ and $\sf X$($\sf I$) : = $\sf X$ are called, respectively, a Macaulay representation and a CM-scheleton of $\sf I$ : = $\mathfrak$P($\mathbb {L}$); each λ = $\ell_{{\upsilon{\sf a}}}^{}$λ$\scriptstyle \sf a$∈$\mathbb {L}$ is called a CM-functional and each $\sf x$ = ($\sf a$, υ)∈$\sf X$ a CM-card.

If, moreover, for each λ = $\ell_{{\upsilon{\sf a}}}^{}$λ$\scriptstyle \sf a$∈$\mathbb {L}$, λ = M(λ) = M(υ)λ$\scriptstyle \sf a$, then $\sf I$ is called a CM-ideal, $\sf X$ its CM-scheme, and each $\sf x$ = ($\sf a$, υ)∈$\sf X$ a CM-condition.

We need also to consider, for each m < n, the set

$\displaystyle \cal {T}$[1, m] : = $\displaystyle \cal {T}$∩k[X1,…, Xm]  
  = {X1a1 ... Xmam : (a1,…, am)∈$\displaystyle \mathbb {N}$m},  
$\displaystyle \mathbb {M}$[1, m] : = {M(τ) : τ∈$\displaystyle \cal {T}$[1, m]}  

and the projection

πm : kn $\displaystyle \mapsto$ km, πm(x1,…, xn) = (x1,…, xm),

which we freely use to denote also the projections πm : $\cal {T}$ $\simeq$ $\mathbb {N}$n $\mapsto$ $\mathbb {N}$m $\simeq$ $\cal {T}$[1, m], πm(X1α1 ... Xnαn) = X1α1 ... Xmαm, πm : $\mathbb {M}$ $\mapsto$ $\mathbb {M}$[1, m], πm(M(τ)) = M(πm(τ)), and πm : kn×$\cal {T}$ $\mapsto$ km×$\cal {T}$[1, m], πm($\sf a$, τ) = (πm($\sf a$), πm(τ)). Also, for a CM-condition $\sf x$ = ($\sf a$, υ)∈km×$\cal {T}$[1, m] we also set πm(λ) : = πm(M(υ)λ$\scriptstyle \sf a$) : = M(πm(υ))λπm($\scriptstyle \sf a$).

Remark that a CM-ideal is radical iff υ = 1∀($\sf a$, υ)∈$\sf X$ iff #$\sf X$ = #$\sf Z$. If this happens, with an abuse of notation, we simply identify each CM-condition $\sf x_{i}^{}$ = ($\sf a_{i}^{}$, 1) and the corresponding CM-functional λi = λ$\scriptstyle \sf a_{i}$ with $\sf a_{i}^{}$.

Let $\sf I$⊂$\cal {P}$ be a CM-ideal, and, using the same notation as above, $\mathbb {L}$ : = {λ1,…, λs} and $\sf X$ : = {$\sf x_{1}^{}$,…,$\sf x_{s}^{}$}⊂kn×$\cal {T}$, $\sf x_{i}^{}$ = ($\sf a_{i}^{}$, υi), $\sf a_{i}^{}$ : = (ai1,…, ain), λi = M(υi)λ$\scriptstyle \sf a_{i}$ a Macaulay representation and a CM-scheme of $\sf I$; let us denote $\bf N$ : = $\bf N$($\sf X$) and Φ : = Φ($\sf X$) the result of Cerlienco–Mureddu Correspondence which satisfies [5,6] $\bf N$ : = $\bf N$($\sf I$).

Since $\bf N$ is an order ideal, $\bf T$ : = $\cal {T}$ $\setminus$ $\bf N$ is a monomial ideal whose minimal basis $\bf G$ : = {$\sf t_{1}^{}$,…,$\sf t_{r}^{}$} will be ordered so that $\sf t_{1}^{}$ < $\sf t_{2}^{}$ < … < $\sf t_{r}^{}$; we also set

$\displaystyle \bf B$ : = $\displaystyle \left(\vphantom{ \{1\} \cup \{X_i\tau : \tau\in {\bf N}\}}\right.${1}∪{Xiτ : τ∈$\displaystyle \bf N$}$\displaystyle \left.\vphantom{ \{1\} \cup \{X_i\tau : \tau\in {\bf N}\}}\right)$ $\displaystyle \setminus$ $\displaystyle \bf N$.

We extend the ordering of $\sf X$ to $\bf N$ = {τ1,…, τs} enumerating it so that τσ = Φ($\sf x_{\sigma}^{}$), for each σ and let us denote the ordering of $\sf X$ and $\bf N$ by $\prec$ so that for each α, β, τα $\prec$ τβ,$\sf x_{\alpha}^{}$ $\prec$ $\sf x_{\beta}^{}$$\iff$α < β.

For a term τ : = X1d1 ... Xndn∈$\cal {T}$ $\setminus$ $\bf N$($\sf X$) such that $\bf N$∪{τ} is an order ideal, we define, for each m, 1≤m≤n:

$\displaystyle \sf N_{{m}}^{}$(τ) : = {ω∈$\displaystyle \cal {T}$[1, m] : τ > ωXm+1dm+1 ... Xndn∈$\displaystyle \bf N$},  
$\displaystyle \sf A_{{m}}^{}$(τ) : = {Φ-1(ωXm+1dm+1 ... Xndn) : ω∈$\displaystyle \sf N_{{m}}^{}$(τ)},  
$\displaystyle \sf B_{{m}}^{}$(τ) : = πm($\displaystyle \sf A_{{m}}^{}$(τ)),  
$\displaystyle \sf C_{{m}}^{}$(τ) : = {πm(λ)∈$\displaystyle \sf B_{{m}}^{}$(τ) : πm-1(λ) $\displaystyle \not\in$$\displaystyle \sf B_{{m-1}}^{}$(τ)},  
$\displaystyle \sf L_{{m}}^{}$(τ) : = {λ∈$\displaystyle \mathbb {L}$ : πm(λ)∈$\displaystyle \sf C_{{m}}^{}$(τ)},  
$\displaystyle \sf D_{{m}}^{}$(τ) : = {$\displaystyle \sf x_{i}^{}$∈$\displaystyle \sf X$ : πm(λi)∈$\displaystyle \sf C_{{m}}^{}$(τ)},  
$\displaystyle \sf M_{{m}}^{}$(τ) : = {ω∈$\displaystyle \cal {T}$[1, m] :  
        ω < Xmdm, ωXm+1dm+1 ... Xndn∈$\displaystyle \bf N$},  
$\displaystyle \mathfrak$Mm(τ) : = {ω∈$\displaystyle \sf M_{m}^{}$(τ) : ω $\displaystyle \prec$ τ},  

where, with slight abuse of notation, we have $\sf N_{{n}}^{}$(τ) : = {ω∈$\cal {T}$ : ω < τ}, $\sf A_{{n}}^{}$(τ) : = {λ : Φ(λ) < τ}, $\sf C_{{1}}^{}$(τ) : = $\sf B_{{1}}^{}$(τ).

Th. [*] below proves, if $\sf I$ is radical, the existence, for each m, 1≤m≤n and each τ : = X1d1 ... Xndn∈$\bf G$ (resp. $\bf N$) of a suitable polynomial gmτ∈Span$\nolimits_{k}^{}$$\left(\vphantom{\{X_m^{d_m}\}\cup{\sf M}_{m}(\tau)}\right.${Xmdm}∪$\sf M_{{m}}^{}$(τ)$\left.\vphantom{\{X_m^{d_m}\}\cup{\sf M}_{m}(\tau)}\right)$ — resp. Span$\nolimits_{k}^{}$$\left(\vphantom{\{X_m^{d_m}\}\cup{{\mathfrak M}}_{m}(\tau)}\right.${Xmdm}∪$\mathfrak$Mm(τ)$\left.\vphantom{\{X_m^{d_m}\}\cup{{\mathfrak M}}_{m}(\tau)}\right)$. In terms of such polynomials we define for each τ∈$\bf N$∪$\bf G$ and each m, 1≤m≤n:

$\displaystyle \sf R_{{m}}^{}$(τ) : = {$\displaystyle \sf b$∈$\displaystyle \sf C_{{m}}^{}$(τ) : $\displaystyle \prod_{{\nu=1}}^{{m-1}}$gντ($\displaystyle \sf b$)≠0},$\displaystyle \cr$$\displaystyle \sf E_{{m}}^{}$(τ)  

Denote for each τ∈$\bf N$
$\displaystyle \mathbb {L}$(τ) : = {λ∈$\displaystyle \mathbb {L}$ : λ $\displaystyle \prec$ Φ-1(τ)} = {λ∈$\displaystyle \mathbb {L}$ : Φ(λ) $\displaystyle \prec$ τ},  
$\displaystyle \mathfrak$X(τ) : = {$\displaystyle \sf x_{j}^{}$ : λj∈$\displaystyle \mathbb {L}$(τ)},  
$\displaystyle \sf I$($\displaystyle \mathbb {L}$(τ)) : = $\displaystyle \mathfrak$P(Span$\displaystyle \nolimits_{k}^{}$($\displaystyle \mathbb {L}$(τ))),  

and, for each τ∈$\bf N$∪$\bf B$:
$\displaystyle \mathfrak$N(τ) : = {ω∈$\displaystyle \bf N$ : ω $\displaystyle \prec$ τ},  
$\displaystyle \mathfrak$Mm(τ) : = {ω∈$\displaystyle \sf M_{m}^{}$ : ω $\displaystyle \prec$ τ}.