ECCC-Report TR24-092https://eccc.weizmann.ac.il/report/2024/092Comments and Revisions published for TR24-092en-usFri, 17 May 2024 08:21:42 +0300
Paper TR24-092
| Hilbert Functions and Low-Degree Randomness Extractors |
Alexander Golovnev,
Zeyu Guo,
Pooya Hatami,
Satyajeet Nagargoje,
Chao Yan
https://eccc.weizmann.ac.il/report/2024/092For $S\subseteq \mathbb{F}^n$, 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.Fri, 17 May 2024 08:21:42 +0300https://eccc.weizmann.ac.il/report/2024/092