CayleyFormula -- Calculate determinant of resultant complex.

Usage:

Res = CayleyFormula(complexRes, mon, choice)
Inputs:
- complexRes, a complex, contains modules over the coefficient ring and the differentials
- mon, a list, exponents of characters which yield bases of its modules
- choice, a list, flags specify admissible sub-modules required for the Cayley formula
Outputs:
- Res, a ring element, resultant evaluated as determinant of resultant complex

Description

A resultant complex must first be determined for this command. Its terms are 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. Finally, a system with two essential sub-families is considered. Even in such cases, the definition of toric resultant complexes and their admissible sub-modules is valid. Evaluating the Cayley formula yields the unit, without checking for exceptions.

i1 : supp = { {{2,0,0},{1,2,2},{0,4,4}}, {{1,0,0},{0,2,2}}, {{0,5,3}}, {{0,0,0},{1,3,2},{4,1,3}} };

i2 : A = QQ[c_1..c_(#flatten supp)];

i3 : (complexRes, mon, choice) = resultantComplex( A, supp, {{0,-4},{-2,-1},{-5},{-13,0,-3}}, {-1/8,1/3,4/7} );

i4 : complexRes

      27      54      27
o4 = A   <-- A   <-- A   <-- 0
                              
     0       1       2       3

o4 : Complex

i5 : Res = CayleyFormula(complexRes, mon, choice)

o5 = 1

o5 : A

For the programmer

The object CayleyFormula is a method function.

The source of this document is in ResultantComplexes.m2:464:0.