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# bezoutianMatrix -- returns a matrix associated to generalized resultants

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

## See also

- eliminationMatrix -- returns a matrix that represents the image of the map
- macaulayFormula -- returns two matrices such that the ratio of their determinants is the Macaulay resultant

## Ways to use bezoutianMatrix:

- `bezoutianMatrix(List,Matrix)`

## For the programmer

The object bezoutianMatrix is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/EliminationMatrices.m2:1234:0.