Resultant complexes of toric systems

Computer aided design and motion control lead to algebraic systems.Resultants of multivariate polynomials are useful to solve such problems.Following Gelfand, Kapranov and Zelevinsky, we calculate them via the Cayley Formula as determinant of a complex formed by global sections of sheaves.These arise from the Koszul complex generated by the polynomials, which we twist by a reflexive rank one bundle corresponding to the shift of Newton polytopes by a rational vector, introduced by Canny and Emiris.Again, inspired by these authors, we apply tight mixed subdivisions of the polytopes to obtain regular minors of the differentials required to evaluate the Cayley formula.Besides the assumption that the Minkowski sum of all Newton polytopes in the system should be full dimensional, there are no further constraints on the set of exponents defining the input polynomials with indeterminate coefficients.Consequently, our resultant coincide with the definition of D'Andrea and Sombra.This complements the package SparseResultant implemented by Staglian (2021) which requires stricter assumptions, including that each individual Newton polytope must be full-dimensional.Let us assume the Minkowski sum Q of system's Newton polytopes is n-dimensional, so that its normal fan determines the complete normal toric variety X as described in [12].The polynomials (1) specify a Koszul complex of sheaves on this variety.It was shown in [20] that after twisting with multiples of any line bundle, its global sections form a complex whose determinant agrees with the resultant.Since the powers are not specified, we use instead a reflexive rank-1-bundle that corresponds to the displacement of Newton polytopes introduced by [6].In addition, they used tight coherent mixed subdivisions of polytopes