eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-16
41:1
41:24
10.4230/LIPIcs.APPROX/RANDOM.2024.41
article
Hilbert Functions and Low-Degree Randomness Extractors
Golovnev, Alexander
1
https://orcid.org/0000-0002-7847-1027
Guo, Zeyu
2
https://orcid.org/0000-0001-7893-4346
Hatami, Pooya
2
https://orcid.org/0000-0001-7928-8008
Nagargoje, Satyajeet
1
https://orcid.org/0009-0003-0452-7360
Yan, Chao
1
https://orcid.org/0000-0001-6482-6643
Georgetown University, Washington, DC, United States of America
The Ohio State University, Columbus, OH, United States of America
For S ⊆ 𝔽ⁿ, consider the linear space of restrictions of degree-d polynomials to S. The Hilbert function of S, denoted h_S(d,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 𝔽ⁿ 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 𝔽_qⁿ, 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol317-approx-random2024/LIPIcs.APPROX-RANDOM.2024.41/LIPIcs.APPROX-RANDOM.2024.41.pdf
Extractors
Dispersers
Circuits
Hilbert Function
Randomness
Low Degree Polynomials