For $S\subseteq \mathbb{F}^n$, consider the linear space of restrictions of degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted $\mathrm{h}_S(d,\mathbb{F})$, is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets $S$ of arbitrary finite grids in $\mathbb{F}^n$ with a fixed size $|S|$. We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size $|S|$. Understanding the smallest values of Hilbert functions is closely related to the study of degree-$d$ closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree-$d$ closures of subsets of $\mathbb{F}_q^n$, which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022). We use the bounds on the Hilbert function and degree-$d$ closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known.

翻译：对于 $S\subseteq \mathbb{F}^n$，考虑次数为 $d$ 的多项式在 $S$ 上的限制所构成的线性空间。$S$ 的希尔伯特函数，记为 $\mathrm{h}_S(d,\mathbb{F})$，是该空间的维数。我们得到了 $\mathbb{F}^n$ 中任意有限网格子集 $S$（固定大小 $|S|$）的希尔伯特函数最小值的紧致下界。我们通过证明该最小值与一个组合量（即大小为 $|S|$ 的下闭集中最少低汉明重量点的个数）相等来实现这一点。理解希尔伯特函数的最小值与集合的 $d$ 次闭包（Nie 和 Wang 在《组合理论杂志》A 辑，2015 年引入的概念）密切相关。我们利用希尔伯特函数的界得到了 $\mathbb{F}_q^n$ 子集的 $d$ 次闭包大小的紧致界，从而回答了 Doron、Ta-Shma 和 Tell 在《计算复杂性》2022 年提出的问题。我们利用希尔伯特函数和集合的 $d$ 次闭包的界证明随机低度多项式是可采样随机性源的提取器。特别地，我们证明了对于由常次数多项式和多项式规模电路生成的源，存在低度提取器和分散器。直到最近，甚至任意确定性提取器对这类源的存在性也尚未可知。