# eliminationMatrix(List,Matrix,Matrix) -- returns a matrix corresponding to a residual resultant

## Synopsis

• Function: eliminationMatrix
• Usage:
eliminationMatrix(v, r, m)
• Inputs:
• v, a list, list v of variables with respect to which the polynomials are homogeneous and from which one wants to remove these variables
• r, , a single row matrix describing the base locus
• m, , a matrix corresponding to the decomposition of a polynomial system over the base locus
• Optional inputs:
• Strategy => ..., default value null, returns a matrix that represents the image of the map
• Outputs:
• , a matrix corresponding to the residual resultant

## Description

If the strategy is 'CM2Residual':

Suppose given a homogeneous ideal locally complete intersection Cohen-Macaulay codimension 2 $J=(g_1,..,g_n)$, such that $I=(f_1,..,f_m)$ is included in J and (I:J) is a residual intersection. Let H be the matrix that I=J.H. Let R be the matrix of the first syzygies of J. This function computes an elimination matrix corresponding to the residual resultant over V(I) over V(J).

 i1 : R = QQ[X,Y,Z,x,y,z] o1 = R o1 : PolynomialRing i2 : F = matrix{{x*y^2,y^3,x*z^2,y^3+z^3}} o2 = | xy2 y3 xz2 y3+z3 | 1 4 o2 : Matrix R <--- R i3 : G = matrix{{y^2,z^2}} o3 = | y2 z2 | 1 2 o3 : Matrix R <--- R i4 : M = matrix{{1,0,0},{0,1,0},{0,0,1},{-X,-Y,-Z}} o4 = | 1 0 0 | | 0 1 0 | | 0 0 1 | | -X -Y -Z | 4 3 o4 : Matrix R <--- R i5 : H = (F//G)*M o5 = {2} | -Xy+x -Yy+y -Zy | {2} | -Xz -Yz -Zz+x | 2 3 o5 : Matrix R <--- R i6 : l = {x,y,z} o6 = {x, y, z} o6 : List i7 : L=eliminationMatrix (l,G,H, Strategy => CM2Residual) o7 = {3} | 0 0 0 0 0 0 1 0 0 0 0 0 | {3} | 0 0 0 0 0 0 -X 1 0 -Y+1 0 0 | {3} | 0 0 -Y 0 0 0 -Z 0 1 0 0 0 | {3} | -1 0 0 0 0 0 0 -X 0 0 -Y+1 0 | {3} | 0 0 X -Y 0 0 0 -Z -X -Z 0 -Y+1 | {3} | 0 0 0 0 -Y -1 0 0 -Z 0 0 0 | {3} | X Y-1 0 0 0 Z 0 0 0 0 0 0 | {3} | 0 0 0 X 0 0 0 0 0 0 -Z 0 | {3} | 0 0 0 0 X 0 0 0 0 0 0 -Z | {3} | X Y 0 0 0 Z 0 0 0 0 0 0 | 10 12 o7 : Matrix R <--- R i8 : maxCol L o8 = {{3} | 0 0 0 0 0 0 1 0 0 0 |, {0, 1, 2, 3, 4, 5, 6, 7, 8, {3} | 0 0 0 0 0 0 -X 1 0 -Y+1 | {3} | 0 0 -Y 0 0 0 -Z 0 1 0 | {3} | -1 0 0 0 0 0 0 -X 0 0 | {3} | 0 0 X -Y 0 0 0 -Z -X -Z | {3} | 0 0 0 0 -Y -1 0 0 -Z 0 | {3} | X Y-1 0 0 0 Z 0 0 0 0 | {3} | 0 0 0 X 0 0 0 0 0 0 | {3} | 0 0 0 0 X 0 0 0 0 0 | {3} | X Y 0 0 0 Z 0 0 0 0 | ------------------------------------------------------------------------ 9}} o8 : List

If the strategy is 'ciResidual':

This function basically computes the matrix of the first application in the resolution of (I:J) given in the article of Bruns, Kustin and Miller: 'The resolution of the generic residual intersection of a complete intersection', Journal of Algebra 128.

The first argument is a list of homogeneous polynomials $J=(g_1,..,g_n)$forming a complete intersection with respect to the variables 'varList'. Given a system of homogeneous $I=(f_1,..,f_m)$, such that I is included in J and (I:J) is a residual intersection, one wants to compute a sort of resultant of (I:J). The second argument is the matrix M such that I=J.M. The output is a generically (with respect to the other variables than v) surjective matrix such that the determinant of a maximal minor is a multiple of the resultant of I on the closure of the complementary of V(J) in V(I). Such a minor can be obtain with maxMinor.

 i9 : R=QQ[a_0,a_1,a_2,a_3,a_4,b_0,b_1,b_2,b_3,b_4,c_0,c_1,c_2,c_3,c_4,x,y,z] o9 = R o9 : PolynomialRing i10 : G=matrix{{z,x^2+y^2}} o10 = | z x2+y2 | 1 2 o10 : Matrix R <--- R i11 : H=matrix{{a_0*z+a_1*x+a_2*y,b_0*z+b_1*x+b_2*y,c_0*z+c_1*x+c_2*y},{a_3,b_3,c_3}} o11 = | a_1x+a_2y+a_0z b_1x+b_2y+b_0z c_1x+c_2y+c_0z | | a_3 b_3 c_3 | 2 3 o11 : Matrix R <--- R i12 : L=eliminationMatrix ({x,y,z},G,H, Strategy => ciResidual) o12 = {2} | a_3 b_3 c_3 -a_3b_1+a_1b_3 0 0 {2} | 0 0 0 -a_3b_2+a_2b_3 -a_3b_1+a_1b_3 0 {2} | a_1 b_1 c_1 -a_3b_0+a_0b_3 0 -a_3b_1+a_1b_3 {2} | a_3 b_3 c_3 0 -a_3b_2+a_2b_3 0 {2} | a_2 b_2 c_2 0 -a_3b_0+a_0b_3 -a_3b_2+a_2b_3 {2} | a_0 b_0 c_0 0 0 -a_3b_0+a_0b_3 ----------------------------------------------------------------------- -a_3c_1+a_1c_3 0 0 -b_3c_1+b_1c_3 -a_3c_2+a_2c_3 -a_3c_1+a_1c_3 0 -b_3c_2+b_2c_3 -a_3c_0+a_0c_3 0 -a_3c_1+a_1c_3 -b_3c_0+b_0c_3 0 -a_3c_2+a_2c_3 0 0 0 -a_3c_0+a_0c_3 -a_3c_2+a_2c_3 0 0 0 -a_3c_0+a_0c_3 0 ----------------------------------------------------------------------- 0 0 | -b_3c_1+b_1c_3 0 | 0 -b_3c_1+b_1c_3 | -b_3c_2+b_2c_3 0 | -b_3c_0+b_0c_3 -b_3c_2+b_2c_3 | 0 -b_3c_0+b_0c_3 | 6 12 o12 : Matrix R <--- R

If the strategy is 'byResolution':

This function computes the matrix of the first application in the resolution of (I:J) given by resolutionin degree regularity

 i13 : R=QQ[a_0,a_1,a_2,a_3,a_4,b_0,b_1,b_2,b_3,b_4,c_0,c_1,c_2,c_3,c_4,x,y,z] o13 = R o13 : PolynomialRing i14 : G=matrix{{z,x^2+y^2}} o14 = | z x2+y2 | 1 2 o14 : Matrix R <--- R i15 : H=matrix{{a_0*z+a_1*x+a_2*y,b_0*z+b_1*x+b_2*y,c_0*z+c_1*x+c_2*y},{a_3,b_3,c_3}} o15 = | a_1x+a_2y+a_0z b_1x+b_2y+b_0z c_1x+c_2y+c_0z | | a_3 b_3 c_3 | 2 3 o15 : Matrix R <--- R i16 : L=eliminationMatrix ({x,y,z},G,H, Strategy => byResolution) o16 = {2} | a_3b_1-a_1b_3 0 0 a_3c_1-a_1c_3 {2} | a_3b_2-a_2b_3 a_3b_1-a_1b_3 0 a_3c_2-a_2c_3 {2} | a_3b_0-a_0b_3 0 a_3b_1-a_1b_3 a_3c_0-a_0c_3 {2} | 0 a_3b_2-a_2b_3 0 0 {2} | 0 a_3b_0-a_0b_3 a_3b_2-a_2b_3 0 {2} | 0 0 a_3b_0-a_0b_3 0 ----------------------------------------------------------------------- 0 0 b_3c_1-b_1c_3 0 0 a_3c_1-a_1c_3 0 b_3c_2-b_2c_3 b_3c_1-b_1c_3 0 0 a_3c_1-a_1c_3 b_3c_0-b_0c_3 0 b_3c_1-b_1c_3 a_3c_2-a_2c_3 0 0 b_3c_2-b_2c_3 0 a_3c_0-a_0c_3 a_3c_2-a_2c_3 0 b_3c_0-b_0c_3 b_3c_2-b_2c_3 0 a_3c_0-a_0c_3 0 0 b_3c_0-b_0c_3 ----------------------------------------------------------------------- a_3 b_3 c_3 | 0 0 0 | a_1 b_1 c_1 | a_3 b_3 c_3 | a_2 b_2 c_2 | a_0 b_0 c_0 | 6 12 o16 : Matrix R <--- R