  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.
