Sparse Nullstellensatz, resultants and determinants of complexes

TL;DR

This paper refines the understanding of sparse polynomial systems by providing a formula for the sparse resultant as a determinant of a Koszul complex, extending previous results on Bezout identities.

Contribution

It introduces a new formula for the sparse resultant using Koszul complexes, generalizing Tuitman's work on supports of Bezout identities.

Findings

Derived a determinant formula for sparse resultants

Extended Tuitman's results on Bezout identities

Applied to systems with one more polynomial than the dimension

Abstract

We refine and extend a result by Tuitman on the supports of a Bezout identity satisfied by a finite sequence of sparse Laurent polynomials without common zeroes in the toric variety associated to their supports. When the number of these polynomials is one more than the dimension of the ambient space, we obtain a formula for computing the sparse resultant as the determinant of a Koszul type complex.