# macaulayFormula -- returns two matrices such that the ratio of their determinants is the Macaulay resultant

## Synopsis

• Usage:
macaulayFormula(v,m)
• Inputs:
• v, a list, a list of n variables such that the polynomials f1,...,fn are homogeneous with respect to these variables
• m, , a single row matrix with polynomials f1,...,fn
• Outputs:
• a list, a list of two matrices such that the ratio of their determinants is the Macaulay resultant of f_1,...,f_n with respect to the variables v

## Description

Let f1,...,fn be a polynomials two groups of variables X1,...,Xn and a1,...,as and such that f1,...,fn are homogeneous polynomials with respect to the variables X1,...,Xn. This function returns two matrices M1 and M2 such that det(D1)/det(D2) is the Macaulay resultant of f1,...,fn providing det(D2) is nonzero.

Remark: if D2 is the empty matrix, its determinant has to be understood as 1 (and not zero, which is the case in Macaulay2 since the empty matrix is identified to the zero.

 ```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 = macaulayFormula (l,M) o7 = ({1} | a d g |, 0) {1} | b e h | {1} | c f i | o7 : Sequence```