# bezoutianMatrix -- returns a matrix associated to generalized resultants

## Synopsis

• Usage:
bezoutianMatrix(v, m)
• Inputs:
• v, a list, a list of n-1 variables to be eliminated form the fi's
• m, , a single row matrix with (affine) polynomials $f_1,...,f_n$
• Outputs:
• , an elimination matrix

## Description

Let R be a polynomial ring in two groups of variables $X_1,...,X_{n-1}$ and $a_1,...,a_s$. The variables $a_1,...,a_s$ are seen as parameters and the variables $X_1,...,X_{n-1}$ are to be eliminated. Being given a row matrix $f_1,...,f_n$ where each $f_i$ is a polynomial in $X_1,...,X_{n-1}$ and $a_1,...,a_s$, this function returns an elimination matrix that only depends on the parameters $a_1,...,a_s$ and whose maximal nonzero minor yields a multiple of the generalized resultant associated to $f_1,...,f_n$

 i1 : R=QQ[a..i,x,y] o1 = R o1 : PolynomialRing i2 : f1 = a*x+b*y+c o2 = a*x + b*y + c o2 : R i3 : f2 = d*x+e*y+f o3 = d*x + e*y + f o3 : R i4 : f3 = g*x+h*y+i o4 = g*x + h*y + i o4 : R i5 : M = matrix{{f1,f2,f3}} o5 = | ax+by+c dx+ey+f gx+hy+i | 1 3 o5 : Matrix R <--- R i6 : l = {x,y} o6 = {x, y} o6 : List i7 : MR = bezoutianMatrix (l,M) o7 = | -ceg+bfg+cdh-afh-bdi+aei | 1 1 o7 : Matrix R <--- R