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
|