Macaulay: The u-resultant.

Let us consider r≤n homogeneous polynomials

#math355#

#tex2html_wrap_indisplay11681# : = {f1,…, fr}⊂k[x1,…, xn]

of degrees #math356#d1≤ ... ≤dr.

One can therefore expect that the ideal #math357#M : = (f1,…, fr) has rank r, in which case Macaulay considersits extension/contraction ideal

#math358#

M(r) : = Mk(xr+1,…, xn)[x1,…, xr]∩k[x1,…, xn]

and is aware that, if a 'generic' change of coordinates has been already performed, each #math359#fi∈M(r) is homogeneous of degree di in the variables#math360#x1,…, xr and the assumption on the rank is satisfied if and only if the resultant of the r fis w.r.t. the r - 1 variables #math361#x1,…, xr-1, #math362#F#tex2html_wrap_inline11697#∈k[xr+1,…, xn][xr] does not vanish, thus granting the existence of a root.

In this context and under these assumptions, adapting the notation 49 of Chapters~31-32 and 39 we set

#math363#

k[x1,…, xn] = k[Z1,…, Zr, V1,…, Vd] = k[xr+1,…, xn][x1,…, xr],

denote#math364#π : k[x1,…, xn] = k[V1,…, Vd][Z1,…, Zr]→k[Z1,…, Zr] the projection defined by #math365#π(F) = F(Z1,…, Zr, 0,…, 0], for each 50

#math366#

F(x1,…, xr, xr+1,…, xn) = F(Z1,…, Zr, V1,…, Vd),

#math367#K = k(V1,…, Vd), #math368##tex2html_wrap_inline11705#R = k[V1,…, Vd], and consider

#math369#

(f1,…, fr) = M(r)⊂#tex2html_wrap_indisplay11707#R[Z1,…, Zr],

remarking that with this new notation we have #math370#F#tex2html_wrap_inline11709#∈#tex2html_wrap_inline11710#R[Zr].

We begin by remarking that the polynomials

#math371#

#tex2html_wrap_indisplay11712# : = fi(Z0V1,…, Z0Vd, Z1,…, Zr-1, Z0Zr)∈#tex2html_wrap_indisplay11713#R[Z0, Z1,…, Zr-1, Zr],

are homogeneous in the variables #math372#Z1,…, Zr-1, Z0 and that #math373#F#tex2html_wrap_inline11716#∈#tex2html_wrap_inline11717#R[Zr] is the resultant #math374#F#tex2html_wrap_inline11719# w.r.t. #math375#Z1,…, Zr-1, Z0 of #math376##tex2html_wrap_inline11722# : = {#tex2html_wrap_inline11723#,…,#tex2html_wrap_inline11724#}; moreover F#tex2html_wrap_inline11726# is a homogeneous polynomial in the variables #math377#V1,…, Vd, Zr of degree #math378#D : = #tex2html_wrap_inline11729#di so that #math379##tex2html_wrap_inline11731#(F#tex2html_wrap_inline11732#) = Rr+1ZrD where (by the assumption on the rank)

#math380#

Rr+1 = #tex2html_wrap_indisplay11734##tex2html_wrap_indisplay11735#π(f1),…, π(fr)#tex2html_wrap_indisplay11736#≠0.

Instead of solving for one of the unknown variables Zi, we solve for their <#1794#>Liouville substitution<#1794#>

#math381#

Z = U1Z1 + U2Z2 + ... + UrZr

setting #math382#fu : = Z - U1Z1 - U2Z2 - ... - UrZr, considering the polynomial set #math383##tex2html_wrap_inline11743# : = {f1,…, fr, fu} as a subset of #math384##tex2html_wrap_inline11745#R[U1,…, Ur][Z, Z1,…, Zr] and computing their resultant #math385#F#tex2html_wrap_inline11747#(u)∈#tex2html_wrap_inline11748#R[U1,…, Ur][Z] w.r.t. #math386#Z1,…, Zr.


#Definition1804#

With an argument similar to the one we gave for F#tex2html_wrap_inline11751#, setting

#math387#

#tex2html_wrap_indisplay11753# : = fi(Z0V1,…, Z0Vd, Z1,…, Zr), 1≤i≤r#tex2html_wrap_indisplay11754##tex2html_wrap_indisplay11755# : = Z0Z - #tex2html_wrap_indisplay11756#UiZi

and denoting

#math388#

#tex2html_wrap_indisplay11758# : = {#tex2html_wrap_indisplay11759#,…,#tex2html_wrap_indisplay11760#,#tex2html_wrap_indisplay11761#}⊂#tex2html_wrap_indisplay11762#R[U1,…, Ur][Z0, Z1,…, Zr, Z],

we have that #math389#F(u)#tex2html_wrap_inline11764# is the resultant #math390#F(u)#tex2html_wrap_inline11766# of #math391##tex2html_wrap_inline11768# w.r.t. #math392#Z1,…, Zr, Z0 and, being homogeneous in the variables #math393#V1,…, Vd, Z of degree #math394#D : = #tex2html_wrap_inline11772#di, we have #math395##tex2html_wrap_inline11774#(F(u)#tex2html_wrap_inline11775#) = R'r+1ZD where #math396#R'r+1 = #tex2html_wrap_inline11777##tex2html_wrap_inline11778#π(f1),…, π(fr), π(fu)#tex2html_wrap_inline11779#.

If we consider in the expansion of #math397#F(u)#tex2html_wrap_inline11781# the indeterminate coefficient a, representing the coefieient #math398#c(Z, fu) of Z in fu, degree considerations allow to deduce that #math399#aD | R'r+1 and #math400#R'r+1 = aDRr+1 whence #math401#R'r+1 = Rr+1 since the <#1845#>ansatz<#1845#> evaluates a as #math402#c(Z, fu) = 1.

With the same kind of argument as in Theorem~#41T1#1847>, we can deduce that to each root #math403#α(j)r of #math404#F#tex2html_wrap_inline11793# corresponds a root #math405#(α(j)1,…, α(j)r) of #math406##tex2html_wrap_inline11796#; there are D solutions altogether all being 'finite'since #math409#Rr+1≠ 0. Similarly to each of the D roots z(j) of #math410#F(u)#tex2html_wrap_inline11808# corresponds a root #math411#(β(j)1,…, β(j)r, z(j)) of #math412##tex2html_wrap_inline11811#; clearly, up to a reenumerating we have

#math413#

(α(j)1,…, α(j)r) = (β(j)1,…, β(j)r)

and since #math414##tex2html_wrap_inline11814#(β(j)1,…, β(j)r, z(j)) = 0 we have #math415#z(j) = #tex2html_wrap_inline11816#Uiα(j)i so that

#math416#

F#tex2html_wrap_indisplay11818#(u) = R'r+1#tex2html_wrap_indisplay11819##tex2html_wrap_indisplay11820#Z - #tex2html_wrap_indisplay11821#Uiα(j)i#tex2html_wrap_indisplay11822#.

In conclusion
#Proposition1887#


#Remark1894#

In the generalized setting of Remarks~#41R2#1920> and~#41R2b#1921>, Macaulay's result can be read as follows:
#Proposition1922#