# eliminationMatrix -- returns a matrix that represents the image of the map

## Synopsis

• Usage:
eliminationMatrix(..., Strategy => s)
• Optional inputs:
• Strategy => ..., default value null

## Description

If the strategy 's' is 'Sylvester':

Refer to sylvesterMatrix

If the strategy 's' is 'Macaulay':

Let $f_1,..,f_n$ be a polynomials two groups of variables $X_1,...,X_n$ and $a_1,...,a_s$ and such that $f_1,...,f_n$ are homogeneous polynomials with respect to the variables $X_1,...,X_n$. This function returns a matrix which is generically (in terms of the parameters $a_1,...,a_s$) surjective such that the gcd of its maximal minors is the Macaulay resultant of $f_1,...,f_n$.

 i1 : R=QQ[a_0..a_8,x,y,z] o1 = R o1 : PolynomialRing i2 : f1 = a_0*x+a_1*y+a_2*z o2 = a x + a y + a z 0 1 2 o2 : R i3 : f2 = a_3*x+a_4*y+a_5*z o3 = a x + a y + a z 3 4 5 o3 : R i4 : f3 = a_6*x+a_7*y+a_8*z o4 = a x + a y + a z 6 7 8 o4 : R i5 : M = matrix{{f1,f2,f3}} o5 = | a_0x+a_1y+a_2z a_3x+a_4y+a_5z a_6x+a_7y+a_8z | 1 3 o5 : Matrix R <--- R i6 : l = {x,y,z} o6 = {x, y, z} o6 : List i7 : MR = eliminationMatrix (l,M, Strategy => Macaulay) o7 = {1} | a_0 a_3 a_6 | {1} | a_1 a_4 a_7 | {1} | a_2 a_5 a_8 | 3 3 o7 : Matrix R <--- R

If the strategy 's' is 'determinantal':

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.

 i8 : R=QQ[a_0..a_5,b_0..b_5,x,y] o8 = R o8 : PolynomialRing i9 : 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}} o9 = | 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 o9 : Matrix R <--- R i10 : eliminationMatrix(1,{x,y},M, Strategy => determinantal) o10 = {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 o10 : Matrix R <--- R

If the strategy 's' is 'CM2Residual':

Suppose given a homogeneous ideal locally complete intersection Cohen-Macaulay of codimension 2,$J=(g_1,..,g_n)$, such that $I=(f1,..,fm)$ 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).

 i11 : R = QQ[X,Y,Z,x,y,z] o11 = R o11 : PolynomialRing i12 : F = matrix{{x*y^2,y^3,x*z^2,y^3+z^3}} o12 = | xy2 y3 xz2 y3+z3 | 1 4 o12 : Matrix R <--- R i13 : G = matrix{{y^2,z^2}} o13 = | y2 z2 | 1 2 o13 : Matrix R <--- R i14 : M = matrix{{1,0,0},{0,1,0},{0,0,1},{-X,-Y,-Z}} o14 = | 1 0 0 | | 0 1 0 | | 0 0 1 | | -X -Y -Z | 4 3 o14 : Matrix R <--- R i15 : H = (F//G)*M o15 = {2} | -Xy+x -Yy+y -Zy | {2} | -Xz -Yz -Zz+x | 2 3 o15 : Matrix R <--- R i16 : l = {x,y,z} o16 = {x, y, z} o16 : List i17 : L=eliminationMatrix (l,G,H, Strategy => CM2Residual) o17 = {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 o17 : Matrix R <--- R i18 : maxCol L o18 = {{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}} o18 : List

If the strategy 's' 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 $J=(g_1,..,g_m)$ forming a complete intersection with respect to the variables 'varList'. Given a system of homogeneous $I=(f_1,..,f_n)$ such that I is included in J and (I:J) is a residual intersection, one wants to 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 'varList') 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.

 i19 : 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] o19 = R o19 : PolynomialRing i20 : G=matrix{{z,x^2+y^2}} o20 = | z x2+y2 | 1 2 o20 : Matrix R <--- R i21 : 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}} o21 = | 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 o21 : Matrix R <--- R i22 : L=eliminationMatrix ({x,y,z},G,H, Strategy => ciResidual) o22 = {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 o22 : 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

 i23 : 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] o23 = R o23 : PolynomialRing i24 : G=matrix{{z,x^2+y^2}} o24 = | z x2+y2 | 1 2 o24 : Matrix R <--- R i25 : 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}} o25 = | 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 o25 : Matrix R <--- R i26 : L=eliminationMatrix ({x,y,z},G,H, Strategy => byResolution) o26 = {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 o26 : Matrix R <--- R