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 ? 1 $k\geqslant 1$ n $n$ large with respect to $k$ , what is minimum possible degree a polynomial P ? R [ x ? ] $P\in \mathbb {R}[x_1,\dots ,x_n]$ ( 0 ) ? $P(0,\dots ,0)\ne 0$ such that $P$ has zeroes at least all points in { } ? $\lbrace 0,1\rbrace ^n\setminus \lbrace (0,\dots ,0)\rbrace$ ? For = $k=1$ classical theorem Alon Füredi states equals . In this paper, solve problem 2 2$ proving answer + ? 3 $n+2k-3$ As an application, improve result Clifton Huang on configurations hyperplanes $\mathbb {R}^n$ each point covered hyperplanes, but $(0,\dots ,0)$ uncovered. Surprisingly, proof our involves Catalan numbers arguments from enumerative combinatorics.