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 based on what today we could call a solution via linear syzygies.

He assumes to have m₁ variables connected by m₂ linear equations, not being all independent, but connected by m₃ linear equations, again not necessarily linearly independent; we thus obtain s matrices #math611#M_σ = #tex2html_wrap_inline12752#a^(σ)_ij#tex2html_wrap_inline12753#, the i^th matrix having m_σ columns and #math612#m_σ+1 rows, the m_σs being related by #math613##tex2html_wrap_inline12759#(- 1)^σm_σ = 0:

the number of quantities [m₁] will be equal to the number of really independent equations connecting them, and we may obtain by the elimination of these quantities a result #math614#Δ = 0.

The approach, denoting #math615#μ_{#tex2html_wrap_inline12763#} : = #tex2html_wrap_inline12764#(- 1)^{σ-#tex2html_wrap_inline12765#}m_σ = 0, consists in

selecting #math616#μ_s+1 = m_s+1 indexes #math617#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 #math618#μ_s = m_s - μ_s+1 indexes #math619#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 #math620#{1,…, m_s} - I_s
...
selecting #math621#μ_{#tex2html_wrap_inline12780#} = m_{#tex2html_wrap_inline12781#} - μ_{#tex2html_wrap_inline12782#+1} indexes #math622#I_{#tex2html_wrap_inline12784#-1}⊂{1,…, m_{#tex2html_wrap_inline12785#-1}} and computing the determinant #math623#Q_{#tex2html_wrap_inline12787#-1} of the #math624#μ_{#tex2html_wrap_inline12789#}--square minor of #math625#M_{#tex2html_wrap_inline12791#-1} obtained by selecting the rows indexed by #math626#I_{#tex2html_wrap_inline12793#-1} and the columns indexed by #math627#{1,…, m_{#tex2html_wrap_inline12795#}} - I_{#tex2html_wrap_inline12796#}
...
selecting #math628#μ₃ = m₃ - μ₄ indexes #math629#I₂⊂{1,…, m₂} and computing the determinant Q₂ of the μ₂--square minor of M₂ obtained by selecting the rows indexed by I₂ and the columns indexed by #math630#{1,…, m₃} - I₃
computing, on the basis of the remark that #math631#m₁ = μ₂ = m₂ - μ₃, the determinant Q₁ of the μ₂--square minor of M₁ obtained by selecting the columns indexed by #math632#{1,…, m₂} - I₂;

finally, if each Q_i is not zero, one obtaind Δ by computing

#math633#

Δ = Q₁Q₂^-1Q₃Q₄^-1^... = #tex2html_wrap_indisplay12812#Q_σ^{(-1)^σ-1}.

The application considers a set of forms #math634#{f₁,…, f_u} and, fixed a proper degree #math635#d∈#tex2html_wrap_inline12815#N intends to eliminate all terms of degree d among the equations F = 0 where F runs among the forms in the set

#math636#

#tex2html_wrap_indisplay12820# : = {τf_i, 1≤i≤u, τ∈#tex2html_wrap_indisplay12821#T, deg(τf_i) = d};

it consists in computing a linear resolution of the elements in #tex2html_wrap_inline12823# and applying on it the computation suggested above. Cayley however remarks that

I am not in possession of any method of arriving <#3308#>at once<#3308#> 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 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

#math637#

φ(X₁, X₂)	=	#tex2html_wrap_indisplay12829##tex2html_wrap_indisplay12830#A_rsX₁^rX₂^s,
ψ(X₁, X₂)	=	#tex2html_wrap_indisplay12834##tex2html_wrap_indisplay12835#B_rsX₁^rX₂^s,
χ(X₁, X₂)	=	#tex2html_wrap_indisplay12839##tex2html_wrap_indisplay12840#C_rsX₁^rX₂^s

Dixon considers the determinant

#math638#

Δ : = #tex2html_wrap_indisplay12842##tex2html_wrap_indisplay12843##tex2html_wrap_indisplay12844#

and, remarking that it vanishes if we put X₁ = Y₁ and also if we put X₂ = Y₂, and so it is divisible by #math639#(X₁ - Y₁)(X₂ - Y₂), he considers the polynomial

#math640#

D(X₁, X₂, Y₁, Y₂) = #tex2html_wrap_indisplay12849#

which is of degree #math641##tex2html_wrap_inline12851# so that

equating to zero the cofficients of #math642#[Y₁^rY₂^s], for all values of r and s, [D = 0] is equivalent to 2mn equations in [X₁, X₂] and the number of terms in these equations is also 2mn. Thus the eliminant 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 #math643#Δ : = #tex2html_wrap_inline12863##tex2html_wrap_inline12864##tex2html_wrap_inline12865#

In other words, denoting #math644##tex2html_wrap_inline12867#a : = {X₁^rX₂^s;r ;SPMlt; 2n, s ;SPMlt; m}, and #math645##tex2html_wrap_inline12869#b : = {Y₁^rY₂^s;r ;SPMlt; n, s ;SPMlt; 2m}, we have

#math646#

D(X₁, X₂, Y₁, Y₂) = #tex2html_wrap_indisplay12871##tex2html_wrap_indisplay12872#d_τυτυ,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;d_τυ∈#tex2html_wrap_indisplay12873#Z[A_pq, B_rs, C_tu].

Clearly the vanishing of the determinant of the matrix #math647##tex2html_wrap_inline12875#d_τυ#tex2html_wrap_inline12876# is equivalent of the existence of a common root of #math648#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 #math649#i, 0≤i≤n,

#math650#

g(#tex2html_wrap_indisplay12882#) : = g(Y₁,…, Y_i, X_i+1,…, X_n)#tex2html_wrap_indisplay12883#g(X₁,…, X_n)∈#tex2html_wrap_indisplay12884#P,

so that, in particular #math651#g(#tex2html_wrap_inline12888#) = g(X₁,…, X_n) and #math652#g(#tex2html_wrap_inline12892#) = g(Y₁,…, Y_n).

Given n + 1 polynomials #math653#f₁,…, f_n+1∈k[X₁,…, X_n] each of degree n_i in the variable X_i, one can consider the determinant

#math654#

Δ : = #tex2html_wrap_indisplay12901##tex2html_wrap_indisplay12902##tex2html_wrap_indisplay12903#

(5)

110 which is divisible by #math655##tex2html_wrap_inline12905#(X_i - Y_i) giving a polynomial

#math656#

D(X₁, X₂,…, X_n, Y₁,…, Y_n)

of degree #math657#m_i : = (n + 1 - i)n_i - 1 in X_i and #math658#μ_i : = in_i - 1 in Y_i so that

#math659#

D(X₁, X₂,…, X_n, Y₁,…, Y_n) = #tex2html_wrap_indisplay12912##tex2html_wrap_indisplay12913#d_τυτυ

where #math660##tex2html_wrap_inline12915#a : = {X₁^a₁…X_n^a_n : a_i≤m_i}, and #math661##tex2html_wrap_inline12917#b : = {Y₁^α₁…Y_n^α_n : α_i≤μ_i} and

#math662#

##tex2html_wrap_indisplay12919#a = ##tex2html_wrap_indisplay12920#b = n!#tex2html_wrap_indisplay12921#n_i : = s.

#Definition3397#

#Remark3410#

#Remark3422#

#Remark3424#

In a previous paper Dixon gave another interesting computational approach to evaluate the resultant in terms of Cayley's formula: given two polynomials of the same degree

#math664#

U : = #tex2html_wrap_indisplay12927#a_i+1X^n-i,∈k[X], V : = #tex2html_wrap_indisplay12928#a'_i+1X^n-i,

and denoting #math665##tex2html_wrap_inline12930#C(X, Y) : = #tex2html_wrap_inline12931# he states that
#Lemma3440#

#proof3454#

He then fixes ``two sets of arbitrary quantities'' #math666#x₁,…, x_n and #math667#y₁,…, y_n and states
#Proposition3480#

#proof3491#

#Example3518#