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Covering Hypercube $mB^n$

Zihao Huang, Miao Wang, Suijie Wang

Abstract

We generalize the problem of hyperplane coverings from the Boolean cube to the $m$-fold hypercube $mB^n = \{0,1,\ldots,m\}^n$. Let $f_m(n,k)$ denote the minimum number of hyperplanes such that each point of $mB^n$ is covered at least $k$ times while the origin is uncovered. We derive upper and lower bounds for $f_m(n,k)$, and further determine the exact value: $f_m(n,2) = mn + m$. To achieve this, we establish a version of Sauermann--Wigderson Combinatorial Nullstellensatz for $mB^n$, which enables us to construct polynomials with prescribed vanishing multiplicities.

Covering Hypercube $mB^n$

Abstract

We generalize the problem of hyperplane coverings from the Boolean cube to the -fold hypercube . Let denote the minimum number of hyperplanes such that each point of is covered at least times while the origin is uncovered. We derive upper and lower bounds for , and further determine the exact value: . To achieve this, we establish a version of Sauermann--Wigderson Combinatorial Nullstellensatz for , which enables us to construct polynomials with prescribed vanishing multiplicities.
Paper Structure (8 sections, 23 theorems, 97 equations, 1 figure, 1 table)

This paper contains 8 sections, 23 theorems, 97 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main-1']}
  3. Proof of Theorem \ref{['main-2']}
  4. Proof of Lemma \ref{['c11']}
  5. Proof of Lemma \ref{['p1']}
  6. Proof of Claim \ref{['l33']}
  7. Proof of Claim \ref{['l34']}
  8. Supplementary Constructions

Key Result

Theorem 1.1

For any polynomial $P\in \mathbb{R}\left[x_1,\ldots,x_n\right]$, if $P(\bm{0}) \ne 0$ and $P$ has zeros at all points in $B^n\setminus\{\bm{0}\}$, then $\deg P\ge n$.

Figures (1)

  • Figure 1: Logical structure of proofs.

Theorems & Definitions (47)

  • Theorem 1.1: Alon and Füredi, 1993
  • Theorem 1.2: Clifton and Huang, 2020
  • Theorem 1.3: Sauermann and Wigderson, 2022
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1: Ball and Serra, 2009
  • Theorem 2.2
  • proof
  • proof : Proof of Theorem \ref{['main-1']} \ref{['eq2']}
  • Lemma 3.1
  • ...and 37 more