Structured and Punctured Nullstellens\"atze

TL;DR

This paper generalizes Nullstellensatz results related to polynomials vanishing on structured sets, including punctured grids, by analyzing monomial behavior during polynomial division and extending previous theorems.

Contribution

It provides a unified generalization of existing Nullstellensatz results for grids and punctured grids, introducing new insights into monomial invariance during polynomial division.

Findings

Generalized Nullstellensatz for punctured grids.

Identified monomials unaffected during multivariate polynomial division.

Extended nonzero counting theorem to punctured grids.

Abstract

A Nullstellensatz is a theorem providing information on polynomials that vanish on a certain set: David Hilbert's Nullstellensatz (1893) is a cornerstone of algebraic geometry, and Noga Alon's Combinatorial Nullstellensatz (1999) is a powerful tool in the "Polynomial Method", a technique used in combinatorics. Alon's Theorem excludes that a polynomial vanishing on a grid contains a monomial with certain properties. This theorem has been generalized in several directions, two of which we will consider in detail: Terence Tao and Van H. Vu (2006), Uwe Schauz (2008) and Micha\l{} Laso\'n (2010) exclude more monomials, and recently, Bogdan Nica (2023) improved the result for grids with additional symmetries in their side edges. Simeon Ball and Oriol Serra (2009) incorporated the multiplicity of zeros and gave Nullstellens\"atze for punctured grids, which are sets of the form $X \setminus Y$ with both $X,Y$ grids. We generalize some of these results; in particular, we provide a common generalization to the results of Schauz and Nica. To this end, we establish that during multivariate polynomial division, certain monomials are unaffected. This also allows us to generalize Pete L. Clark's proof of the nonzero counting theorem by Alon and F\"uredi to punctured grids.