About almost covering subsets of the hypercube

TL;DR

This paper demonstrates that a stronger version of a recent hypercube polynomial vanishing result can be directly derived from a classical theorem by Alon and F"uredi, simplifying the proof approach.

Contribution

It shows that the improved hypercube polynomial vanishing theorem by Sziklai and Weiner can be obtained directly from Alon and F"uredi's classical result, avoiding complex methods.

Findings

Stronger version of Sziklai and Weiner's result derived from Alon and F"uredi's theorem

Simplification of proof technique for hypercube polynomial vanishing results

Connection established between recent and classical polynomial vanishing theorems

Abstract

Let $\mathbb{F}$ be a field, and consider the hypercube $\{ 0, 1 \}^{n}$ in $\mathbb{F}^{n}$. Sziklai and Weiner (Journal of Combinatorial Theory, Series A 2022) showed that if a polynomial $P ( X_{1}, \dots, X_{n} ) \in \mathbb{F}[ X_{1}, \dots, X_{n}]$ vanishes on every point of the hypercube $\{0,1\}^{n}$ except those with at most $r$ many ones then the degree of the polynomial will be at least $n-r$. This is a generalization of Alon and F\"uredi's fundamental result (European Journal of Combinatorics 1993) about polynomials vanishing on every point of the hypercube except at the origin (point with all zero coordinates). Sziklai and Weiner proved their interesting result using M\"{o}bius inversion formula and the Zeilberger method for proving binomial equalities. In this short note, we show that a stronger version of Sziklai and Weiner's result can be derived directly from Alon and F\"{u}redi's result.