# eliminationMatrix(List,Matrix) -- returns a matrix associated to the Macaulay resultant

## Synopsis

• Function: eliminationMatrix
• Usage:
eliminationMatrix(v,m)
• Inputs:
• v, a list, a list of n variables such that the polynomials $f_1,...,f_n$ are homogeneous with respect to these variables
• m, , a single row matrix with polynomials $f_1,...,f_n$
• Optional inputs:
• Strategy => ..., default value null, returns a matrix that represents the image of the map
• Outputs:
• , a generically surjective matrix such that the gcd of its maximal minors if the Macaulay resultant of f1,...,fn with respect to the variables 'varList'

## Description

Let $f_1,...,f_n$ be a polynomials two groups of variables $X_1,...,X_n$ and $a_1,...,a_s$ and such that $f_1,...,f_n$ are homogeneous polynomials with respect to the variables $X_1,...,X_n$. This function returns a matrix which is generically (in terms of the parameters $a_1,...,a_s$) surjective such that the gcd of its maximal minors is the Macaulay resultant of $f_1,...,f_n$

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