The procedure uses first a 6x8 submatrix of d2 depending only on the y-variables to normalize the o-matrix in d2. Then we compute the product d1’d2 whose entries define all relations between the known and unknown variables of the set-up. After evaluating the relations which are linear in the unknown variables we see that the entries of the matrix e and p depend only on the entries of the a-matrix. Furthermore, the entries of the n-matrix depend only on the a-, c- and o-variables. Updating the original matrices, we obtain normal forms for d1’ and d2 whose entries depend only on the a-,c-, o- and y-variables. The remaining linear relations are stored in the matrix relLin, whereas the quadratic relations are saved in the matrix relPfaf. These relations are usually Pfaffians of rank 5 or 6.
i1 : kk = ZZ/197; |
i2 : s = "1111"; |
i3 : (A,B,subs0) = globalVariables(kk,s); |
i4 : SR = ring(A); |
i5 : D = complexModuloRegularSequence(SR,s); |
i6 : (d1',d2) = setupGeneralMatrices(D,A,B); |
i7 : (relLin,relPfaf,d1'nor,d2nor) = getRelationsAndNormalForm(d1',d2,subs0); |
i8 : betti relLin, betti relPfaf
0 1 0 1
o8 = (total: 1 42, total: 1 4)
0: 1 . 0: 1 .
1: . . 1: . .
2: . . 2: . .
3: . 12 3: . 4
4: . .
5: . 30
o8 : Sequence
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i9 : transpose relPfaf
o9 = {-4} | a_(3,2,3)a_(3,1,3)-a_(2,2,3)a_(2,1,2)+a_(1,1,3)a_(1,1,2) |
{-4} | -a_(3,2,3)a_(3,0,3)+a_(2,2,3)a_(2,0,2)+a_(0,0,3)a_(0,0,2) |
{-4} | -a_(3,1,3)a_(3,0,3)-a_(1,1,3)a_(1,0,1)+a_(0,0,3)a_(0,0,1) |
{-4} | a_(2,1,2)a_(2,0,2)-a_(1,1,2)a_(1,0,1)+a_(0,0,2)a_(0,0,1) |
4 1
o9 : Matrix (kk[a , a , a , a , a , a , a , a , a , a , a , a , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , o , o , o , o , o , o , o , o , o , o , o , o , x , x , y , y , y , y ]) <--- (kk[a , a , a , a , a , a , a , a , a , a , a , a , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , o , o , o , o , o , o , o , o , o , o , o , o , x , x , y , y , y , y ])
3,2,3 3,1,3 3,0,3 2,2,3 2,1,2 2,0,2 1,1,3 1,1,2 1,0,1 0,0,3 0,0,2 0,0,1 0,0 0,2 0,4 0,6 0,7 1,1 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,5 2,7 3,0 3,1 3,3 3,5 3,7 1,0,0 2,0,1 2,1,2 3,0,0 3,1,0 4,0,1 4,2,1 4,3,3 5,1,2 5,2,2 5,3,3 5,4,3 0 1 0 1 2 3 3,2,3 3,1,3 3,0,3 2,2,3 2,1,2 2,0,2 1,1,3 1,1,2 1,0,1 0,0,3 0,0,2 0,0,1 0,0 0,2 0,4 0,6 0,7 1,1 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,5 2,7 3,0 3,1 3,3 3,5 3,7 1,0,0 2,0,1 2,1,2 3,0,0 3,1,0 4,0,1 4,2,1 4,3,3 5,1,2 5,2,2 5,3,3 5,4,3 0 1 0 1 2 3
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