Sparse effective membership problems via residue currents

TL;DR

This paper introduces residue currents on toric varieties to derive degree bounds for solutions to polynomial ideal membership problems, effectively handling sparse systems by relating bounds to Newton polytope volumes.

Contribution

It develops sparse versions of classical theorems like Max N"other's, Macaulay's, and Kollár's Nullstellensatz using residue currents, advancing the analysis of sparse polynomial systems.

Findings

Derived degree bounds depending on Newton polytope volume

Extended classical theorems to sparse polynomial contexts

Provided bounds well-suited for sparse systems

Abstract

We use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems. Our bounds depend on (the volume of) the Newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems. We present sparse versions of Max N\"other's $AF+BG$ Theorem, Macaulay's Theorem, and Koll\'ar's Effective Nullstellensatz, as well as recent results by Hickel and Andersson-G\"otmark.