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 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