Covering Hypercube $mB^n$

TL;DR

This paper extends known bounds on hyperplane coverings from Boolean cubes to general hypercubes, providing exact values for small multiplicities and bounds for larger ones, using polynomial nullstellensatz and combinatorial techniques.

Contribution

It generalizes the Sauermann–Wigderson Nullstellensatz to the hypercube $mB^n$, determining tight bounds for hyperplane coverings with multiplicities.

Findings

Exact values of $f_m(n, k)$ for $k=1,2$

Sharp lower bounds for polynomial degrees with multiplicity conditions

Bounds for $f_m(n, k)$ when $k \\ge 3$ and $n \\ge k-1$

Abstract

A celebrated result of Alon and F\"{u}redi gives a tight lower bound on the number of hyperplanes required to cover all points of the Boolean cube $B^n$ except the origin $\bm{0}$. Recent breakthroughs by Sauermann and Wigderson generalized this to the case where all points of $B^n \setminus \{\mathbf{0}\}$ are covered with multiplicities at least $k$. In this paper, we further extend their result by replacing the Boolean cube with the general hypercube $mB^n = \{0, 1, \dots, m\}^n$. \vspace{2mm} Let $f_m(n, k)$ denote the minimum number of hyperplanes required to cover every point of $mB^n \setminus \{\mathbf{0}\}$ at least $k$ times while leaving the origin uncovered. Our primary contribution is a sharp extension of the Sauermann--Wigderson Combinatorial Nullstellensatz to the setting of $mB^n$. We determine a tight lower bound for the degree of polynomials that vanish with multiplicity at least $k$ at all points of $mB^n \setminus \{\mathbf{0}\}$ and have multiplicity less than $k$ at the origin. As an application, we establish the exact values $f_m(n, k)$ for $k=1,2$ and provide upper and lower bounds for $f_m(n, k)$ when $k \ge 3$ and $n\ge k-1$. The proofs involve a new construction of hyperplanes and a surprisingly elegant application of the Lagrange inversion formula in enumerative combinatorics.