A Gr\"{o}bner Basis Approach to Combinatorial Nullstellensatz

Abstract

In this paper, using some conditions that arise naturally in Alon's combinatorial Nullstellensatz as well as its various extensions and generalizations, we characterize Gr\"{o}bner bases consisting of monic polynomials, which helps us to establish a Nullstellensatz from a Gr\"{o}bner basis perspective. As corollaries of this general Nullstellensatz, we establish four special Nullstellensatz, which, among others, include a common generalization of the Nullstellensatz for multisets established in K\'{o}s, R\'{o}nyai and M\'{e}sz\'{a}ros \cite{23,24} and the Nullstellensatz with multiplicity established in Ball and Serra \cite{9}, and include a punctured Nullstellensatz, generalizing several existing results in the literature. As applications of our punctured Nullstellensatz, we extend some results on hyperplane covering in \cite{9,23,24} to wider settings, and give an alternative proof of the generalized Alon-F\"{u}redi theorem established in Bishnoi, Clark, Potukuchi and Schmitt \cite{12}. Unless specified otherwise, all our results are established over an arbitrary commutative ring $R$.