A Robust Version of Heged\H{u}s's Lemma, with Applications

TL;DR

This paper introduces a robust variant of Hegedüs's lemma for polynomials over finite fields, extending its applicability to cases where polynomials nearly vanish on certain Hamming weight layers, and demonstrates its usefulness through three applications.

Contribution

The paper formulates and proves a robust version of Hegedüs's lemma, broadening its scope to polynomials that almost vanish on specific Hamming weight layers.

Findings

Proved the robust version of Hegedüs's lemma.

Applied the lemma to three different problems in combinatorics and computational complexity.

Extended the utility of the original lemma to approximate vanishing scenarios.

Abstract

Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of some fixed Hamming weight $k\in [q,n-q]$ must also vanish at all points in $\{0,1\}^n$ of weight $k + q$. This lemma was used by Heged\H{u}s (2009) to give a solution to \emph{Galvin's problem}, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrube\v{s}, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-$2$ threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Heged\H{u}s's lemma. Informally, this version says that if a polynomial of degree $o(q)$ vanishes at most points of weight $k$, then it vanishes at many points of weight $k+q$. We prove this lemma and give three different applications.