Polynomials that vanish to high order on most of the hypercube

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 Füredi 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.

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