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Resultants :: ciRes

ciRes -- returns a matrix corresponding to the residual resultant over a complete intersection

Synopsis

Description

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=(g1,..,gm) forming a complete intersection with respect to the variables 'varList'. Given a system of homogeneous I=(f1,..,fn), 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 and maxMinorNum.

i1 : 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]

o1 = R

o1 : PolynomialRing
i2 : G=matrix{{z,x^2+y^2}}

o2 = | z x2+y2 |

             1       2
o2 : Matrix R  <--- R
i3 : 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}}

o3 = | 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
o3 : Matrix R  <--- R
i4 : L=ciRes(G,H,{x,y,z})

o4 = {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
o4 : Matrix R  <--- R
i5 : maxCol L

o5 = {{2} | a_3 b_3 c_3 -a_3b_1+a_1b_3 0              -a_3c_1+a_1c_3 |, {0,
      {2} | 0   0   0   -a_3b_2+a_2b_3 -a_3b_1+a_1b_3 -a_3c_2+a_2c_3 |
      {2} | a_1 b_1 c_1 -a_3b_0+a_0b_3 0              -a_3c_0+a_0c_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 0              |
      {2} | a_0 b_0 c_0 0              0              0              |
     ------------------------------------------------------------------------
     1, 2, 3, 4, 6}}

o5 : List

See also