Proof of the combinatorial nullstellensatz over integral domains in the spirit of Kouba

TL;DR

This paper provides a direct proof of the combinatorial nullstellensatz over integral domains, removing the need for duality theory and using elementary linear algebra tools, thus generalizing the result.

Contribution

It offers a new, more straightforward proof of the nullstellensatz over integral domains that avoids duality theory, broadening its applicability.

Findings

Proof relies on Cramer's rule and Vandermonde's determinant

The nullstellensatz holds over any integral domain

The approach simplifies and generalizes previous proofs

Abstract

It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba (O. Kouba, A duality based proof of the Combinatorial Nullstellensatz. Electron. J. Combin. 16 (2009), #N9) one obtains a direct proof of the nonvanishing-version of Alon's Combinatorial Nullstellensatz for polynomials over an arbitrary integral domain. The proof relies on Cramer's rule and Vandermonde's determinant to explicitly describe a map used by Kouba in terms of cofactors of a certain matrix. That the Combinatorial Nullstellensatz is true over integral domains is a well-known fact which is already contained in Alon's work and emphasized in recent articles of Michalek and Schauz; the sole purpose of the present note is to point out that not only is it not necessary to invoke duality of vector spaces, but by not doing so one easily obtains a more general result.