CannyEmirisCoef -- determine rank one bundle to twist Koszul complex

Usage:

coefsDiv = CannyEmirisCoef( facetPolytope, shiftPolytope )
Inputs:
- facetPolytope, a list, specifies facets of polytope
- shiftPolytope, a matrix, vector with rational coefficients move polytope
Outputs:
- coefsDiv, a list, coefficients of a divisor equivalent to translation of polytope

Description

Following Gelfand, Kapranov and Zelevinsky, we calculate resultants via Cayley Formula as determinant of a complex formed by global sections of a Koszul complex of sheafs CayleyFormula and resultantComplex. To ensure that higher sheaf cohomologies vanish, we twist this complex by a reflexive rank one bundle, which corresponds to the shift of Newton polytopes by a rational vector, introduced by Canny and Emiris. We cannot assume the Weil divisor corresponding to this bundle being Q-Cartier.

i1 : needsPackage "NormalToricVarieties";

i2 : supp = {{1,1,3},{0,2,1},{2,0,4},{2,5,0},{3,2,7}};

i3 : P = convexHull transpose matrix supp;

i4 : X = normalToricVariety P;

i5 : a = CannyEmirisCoef( facets P, matrix {{-2/7},{1/3},{3/5}} )

o5 = {-1, -11, -11, 1, 2, 5}

o5 : List

i6 : D = sum( #rays X, i -> a_i * X_i );

o6 : ToricDivisor on X

i7 : isQQCartier D

o7 = false

Ways to use CannyEmirisCoef:

CannyEmirisCoef(Sequence,Matrix) (missing documentation)

For the programmer

The object CannyEmirisCoef is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/ResultantComplexes.m2:506:0.