eliminationMatrix(ZZ,List,Matrix) -- returns a matrix corresponding to the determinantal resultant, in particular the Macaulay resultant

Synopsis

Function: eliminationMatrix
Usage:

eliminationMatrix(r, v, m)
Inputs:
- r, an integer, corresponding to the regularity index used to form the determinantal resultant
- v, a list, a list of homogeneous variables that are to be eliminated
- m, a matrix, a polynomial matrix
Optional inputs:
- Strategy => ..., default value null, returns a matrix that represents the image of the map
Outputs:
- M, a matrix, a matrix corresponding to the determinantal resultant

Description

Compute the determinantal resultant of an (n,m)-matrix (n<m) of homogeneous polynomials over the projective space of dimension (m-r)(n-r), i.e. a condition on the parameters of these polynomials to have rank(M)<r+1.M

i1 : R=QQ[a_0..a_5,b_0..b_5,x,y]

o1 = R

o1 : PolynomialRing

i2 : M:=matrix{{a_0*x+a_1*y,a_2*x+a_3*y,a_4*x+a_5*y},{b_0*x+b_1*y,b_2*x+b_3*y,b_4*x+b_5*y}}

o2 = | a_0x+a_1y a_2x+a_3y a_4x+a_5y |
     | b_0x+b_1y b_2x+b_3y b_4x+b_5y |

             2      3
o2 : Matrix R  <-- R

i3 : eliminationMatrix(1,{x,y},M, Strategy => determinantal)

o3 = {2} | -a_2b_0+a_0b_2               -a_4b_0+a_0b_4              
     {2} | -a_3b_0-a_2b_1+a_1b_2+a_0b_3 -a_5b_0-a_4b_1+a_1b_4+a_0b_5
     {2} | -a_3b_1+a_1b_3               -a_5b_1+a_1b_5              
     ------------------------------------------------------------------------
     -a_4b_2+a_2b_4               |
     -a_5b_2-a_4b_3+a_3b_4+a_2b_5 |
     -a_5b_3+a_3b_5               |

             3      3
o3 : Matrix R  <-- R

Ways to use this method: