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About almost covering subsets of the hypercube

Arijit Ghosh, Chandrima Kayal, Soumi Nandi

TL;DR

The paper identifies a general lower bound for the degree of a polynomial over a field that vanishes on a subset of the Boolean hypercube but is nonzero at some outside point: $\deg(P) \ge \max\{w, n-W\}$, where $w$ and $W$ are the extreme Hamming weights of points where $P$ does not vanish. It derives this bound directly from the Alon–Füredi result via a reduction that reduces to a lower-dimensional instance, and then obtains a complementary bound using the complement polynomial. It also extends the framework to real polynomials with multiplicity: $\deg(P) \ge \max\{w_t(P), n-W_t(P)\} + 2t - 3$, leveraging the Sauermann–Wigderson generalization, and provides tightness constructions (via Kummer’s theorem and related arguments) to show the bounds are optimal. The results unify and strengthen prior bounds (SW23, Hegedüs) and apply to non-symmetric vanishing sets, broadening the scope of degree lower bounds for Boolean function representations by polynomials.

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 Füredi'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 Möbius 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 Füredi's result.

About almost covering subsets of the hypercube

TL;DR

The paper identifies a general lower bound for the degree of a polynomial over a field that vanishes on a subset of the Boolean hypercube but is nonzero at some outside point: , where and are the extreme Hamming weights of points where does not vanish. It derives this bound directly from the Alon–Füredi result via a reduction that reduces to a lower-dimensional instance, and then obtains a complementary bound using the complement polynomial. It also extends the framework to real polynomials with multiplicity: , leveraging the Sauermann–Wigderson generalization, and provides tightness constructions (via Kummer’s theorem and related arguments) to show the bounds are optimal. The results unify and strengthen prior bounds (SW23, Hegedüs) and apply to non-symmetric vanishing sets, broadening the scope of degree lower bounds for Boolean function representations by polynomials.

Abstract

Let be a field, and consider the hypercube in . Sziklai and Weiner (Journal of Combinatorial Theory, Series A 2022) showed that if a polynomial vanishes on every point of the hypercube except those with at most many ones then the degree of the polynomial will be at least . This is a generalization of Alon and Füredi'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 Möbius 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 Füredi's result.
Paper Structure (4 sections, 9 theorems, 6 equations)

This paper contains 4 sections, 9 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Covering with polynomials from $\mathbb{F}[X_1, \dots, X_n]$
  4. Covering with multiplicities when $\mathbb{F} = \mathbb{R}$

Key Result

Theorem 1

Suppose $P(X_{1}, \dots, X_{n})$ is a polynomial in $\mathbb{F} \left[ X_{1}, \dots, X_{n} \right]$ such that $P(u)=0$ for all $u\in\left\{ 0,1 \right\}^n\setminus\{(0,\dots,0)\}$ and $P(0,\dots,0)\neq 0$. Then $deg(P) \geq n$.

Theorems & Definitions (16)

  • Theorem 1: Alon and Füredi AF93
  • Theorem 2: Sziklai and Weiner SziklaiW23
  • Theorem 3: Hegedüs hegedus2024coversfinitesetspoints
  • Theorem 4: Main result
  • proof : Proof of Theorem \ref{['thm-new-lower-bound']}
  • Corollary 5
  • proof
  • Remark 6
  • Proposition 7
  • Theorem 8: Kummer Kummer1852
  • ...and 6 more