Residues and Resultants

TL;DR

This paper explores the relationships between residues, resultants, and Jacobians within toric geometry, providing formulas that connect these invariants and offering new determinantal expressions for resultants.

Contribution

It establishes denominator formulas for toric and global residues using sparse resultants and introduces a determinantal formula for resultants based on Jacobians.

Findings

Derived denominator formulas for residues in terms of sparse resultants.

Provided a determinantal formula for resultants involving Jacobians.

Connected local and global residues through toric geometry frameworks.

Abstract

Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$ variables. Cox introduced the related notion of the toric residue relative to $n+1$ divisors on an $n$-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formula for resultants based on Jacobians.