Some variants of Macaulay's and Max Noether's theorems

TL;DR

This paper employs residue currents on toric varieties to derive bounds on solutions to polynomial ideal membership problems, especially tailored for sparse polynomial systems based on their Newton polytopes.

Contribution

It introduces variants of Macaulay's and Max Noether's theorems using residue currents, providing bounds suited for sparse polynomial systems.

Findings

Bounds depend on Newton polytopes of polynomials

Results are tailored for sparse polynomial systems

Variants of classical theorems are established

Abstract

We use residue currents on toric varieties to obtain bounds on the support of solutions to polynomial ideal membership problems. Our bounds depend on the Newton polytopes of the polynomial systems and are therefore well adjusted to sparse systems of polynomials. We present variants of classical results due to Macaulay and Max Noether.