Polynomials that vanish to high order on most of the hypercube

TL;DR

This paper determines the minimal degree of polynomials with high-order zeros on most hypercube points, extending classical results and connecting algebraic geometry with combinatorics.

Contribution

It provides a closed-form solution for the minimal degree of such polynomials for all k ≥ 2, generalizing Alon-Füredi's theorem.

Findings

Minimal degree is n+2k-3 for all k≥2.

Improves bounds on hyperplane configurations in real space.

Connects polynomial vanishing properties with Catalan numbers.

Abstract

Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed $k\geq 1$ and $n$ large with respect to $k$, what is the minimum possible degree of a polynomial $P\in \mathbb{R}[x_1,\dots,x_n]$ with $P(0,\dots,0)\neq 0$ such that $P$ has zeroes of multiplicity at least $k$ at all points in $\{0,1\}^n\setminus \{(0,\dots,0)\}$? For $k=1$, a classical theorem of Alon and F\"uredi states that the minimum possible degree of such a polynomial equals $n$. In this paper, we solve the problem for all $k\geq 2$, proving that the answer is $n+2k-3$. As an application, we improve a result of Clifton and Huang on configurations of hyperplanes in $\mathbb{R}^n$ such that each point in $\{0,1\}^n\setminus \{(0,\dots,0)\}$ is covered by at least $k$ hyperplanes, but the point $(0,\dots,0)$ is uncovered. Surprisingly, the proof of our result involves Catalan numbers and arguments from enumerative combinatorics.