Some extensions of Alon's Nullstellensatz

TL;DR

This paper extends Alon's Nullstellensatz by allowing multiple points and generalizing it to arbitrary commutative rings, leading to new combinatorial applications such as hyperplane coverings of discrete cubes.

Contribution

It introduces two significant extensions of Alon's Nullstellensatz, broadening its applicability in combinatorics and algebra.

Findings

Extended the nonvanishing theorem to multiple points

Generalized the theorem to arbitrary commutative rings

Applied the extensions to hyperplane covering problems

Abstract

Alon's combinatorial Nullstellensatz, and in particular the resulting nonvanishing criterion is one of the most powerful algebraic tools in combinatorics, with many important applications. In this paper we extend the nonvanishing theorem in two directions. We prove a version allowing multiple points. Also, we establish a variant which is valid over arbitrary commutative rings, not merely over subrings of fields. As an application, we prove extensions of the theorem of Alon and F\"uredi on hyperplane coverings of discrete cubes.