A Robust Version of Hegedűs's Lemma, with Applications

Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\F$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \F[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of Hamming weight $k\in [q,n-q]$ must also vanish at all points in $\{0,1\}^n$ of weight $k + q$. This lemma was used by Heged\H{u}s (2009) to give a solution to \emph{Galvin's problem}, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrube\v{s}, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-$2$ threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Heged\H{u}s's lemma. Informally, this version says that if a polynomial of degree $o(q)$ vanishes at most points of weight $k$, then it vanishes at many points of weight $k+q$. We prove this lemma and give three different applications.