Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of some fixed 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.

翻译：Hegedűs引理是关于有限域上多项式的如下组合学论断。在特征 $p>0$ 的域 $\mathbb{F}$ 上，设 $q$ 为 $p$ 的幂次，该引理指出：任意次数小于 $q$ 的多线性多项式 $P\in \mathbb{F}[x_1,\ldots,x_n]$，若其在 $\{0,1\}^n$ 中所有固定汉明重量 $k\in [q,n-q]$ 的点上为零，则其也必在 $\{0,1\}^n$ 中所有重量为 $k+q$ 的点上为零。Hegedűs (2009) 利用该引理解决了关于集合系统的极值问题——Galvin问题；Alon、Kumar和Volk (2018) 利用其改进了最佳已知多线性电路下界；Hrubeš、Ramamoorthy、Rao和Yehudayoff (2019) 利用其证明了计算某些对称函数时深度为2的阈值电路的最优下界。本文提出Hegedűs引理的一个稳健版本。非形式化地讲，该版本指出：若次数为 $o(q)$ 的多项式在大多数重量为 $k$ 的点上为零，则其在许多重量为 $k+q$ 的点上为零。我们证明该引理并给出三个不同的应用。