This example yields a resultant with a multiple of 7. (Since the family of supports do not satisfy any particular conditions here, the determinant of the complex may have multiplicities. It agrees with the resultant, redefined by D'Andrea and Sombra.) We check the lengths of the co-homologies of the resultant complex localized at the eliminant, which is irreducible and thus generates a prime ideal. Resultant complexes consist of modules over the coefficient ring of the sparse polynomials, they are determined by the lattice points inside sums of Newton polytopes shifted by the Canny Emiris vector, which are stored in mon. The second list choice specifies admissible sub-modules required for the Cayley formula. They are obtained from the lift vectors given in the second argument, provided that these define a tight coherent mixed subdivision (TCMD) of the Newton polytopes:
. To reduce the number of cells in the sub-division, we specify the lift function only at the vertices of the Newton polytopes and not on the entire support sets.
i1 : needsPackage "LocalRings";
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i2 : A = QQ[c_1..c_8];
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i3 : R = A[t_1..t_2];
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i4 : supp = { {{0,0},{2,2},{1,3}}, {{0,2},{2,1},{4,0}}, {{3,0},{1,1}} };
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i5 : f = polynomials( gens A, gens R, supp )
2 2 3 4 2 2 3
o5 = {c t t + c t t + c , c t + c t t + c t , c t + c t t }
2 1 2 3 1 2 1 6 1 5 1 2 4 2 7 1 8 1 2
o5 : List
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i6 : (complexRes,mon,choice) = resultantComplex( f, {{0,-12,-4},{-2,-1},{-3,0}}, {1/8,1/3} );
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i7 : Res = CayleyFormula(complexRes,mon,choice);
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i8 : factor Res
2 2 7
o8 = (c c - c c c + c c )
4 7 5 7 8 6 8
o8 : Expression of class Product
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i9 : CP = complexRes ** A_(radical ideal {Res});
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i10 : for i from 0 to length CP list length HH^i CP
o10 = {7, 0, 0}
o10 : List
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