Combinatorica

, Volume 32, Issue 5, pp 589–605

# Alon’s Nullstellensatz for multisets

• Géza Kós
• Lajos Rónyai
Article

## Abstract

Alon’s combinatorial Nullstellensatz (Theorem 1.1 from [2]) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let $$\mathbb{F}$$ be a field, S 1, S 2,..., S n be finite nonempty subsets of $$\mathbb{F}$$. Alon’s theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set $$S = S_1 \times S_2 \times \ldots \times S_n \subseteq \mathbb{F}^n$$. From this Alon deduces a simple and amazingly widely applicable nonvanishing criterion (Theorem 1.2 in [2]). It provides a sufficient condition for a polynomial f(x 1,..., x n) which guarantees that f is not identically zero on the set S. In this paper we extend these two results from sets of points to multisets. We give two different proofs of the generalized nonvanishing theorem.We extend some of the known applications of the original nonvanishing theorem to a setting allowing multiplicities, including the theorem of Alon and Füredi on the hyperplane coverings of discrete cubes.

## Mathematics Subject Classification (2000)

05-XX 05E40 12D10

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