Hilbert Functions and Low-Degree Randomness Extractors
<|reference_start|>Hilbert Functions and Low-Degree Randomness Extractors: 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.<|reference_end|>
@article{golovnev2024hilbert,
    title={Hilbert Functions and Low-Degree Randomness Extractors},
    author={Alexander Golovnev and Zeyu Guo and Pooya Hatami and Satyajeet
  Nagargoje and Chao Yan},
    journal={arXiv preprint arXiv:2405.10277},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.10277},
    primaryClass={cs.CC}
}