# On degree bounds for the sparse Nullstellensatz

## TL;DR

This paper establishes new degree bounds for the sparse Nullstellensatz and Noether exponent, considering the monomial structure of polynomials, thus improving previous bounds in the sparse setting.

## Contribution

It introduces the first bounds that incorporate the supports of polynomials, advancing the understanding of degree bounds in sparse polynomial ideals.

## Key findings

- Improved upper bounds for degrees in Nullstellensatz
- Bounds that account for polynomial supports
- First bounds considering monomial structure in sparse case

## Abstract

We prove new upper bounds for the degrees in Hilbert's Nullstellensatz and for the Noether exponent of polynomial ideals in terms of the monomial structure of the polynomials involved. Our bounds improve the previously known bounds in the sparse setting and are the first to take into account the different supports of the polynomials.