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Hilbert Functions and Low-Degree Randomness Extractors

Authors Alexander Golovnev , Zeyu Guo , Pooya Hatami , Satyajeet Nagargoje , Chao Yan

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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Extractors
  • Dispersers
  • Circuits
  • Hilbert Function
  • Randomness
  • Low Degree Polynomials

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Abstract

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.

Cite As Get BibTex

Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, and Chao Yan. Hilbert Functions and Low-Degree Randomness Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 41:1-41:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.41

Author Details

Alexander Golovnev
  • Georgetown University, Washington, DC, United States of America
Zeyu Guo
  • The Ohio State University, Columbus, OH, United States of America
Pooya Hatami
  • The Ohio State University, Columbus, OH, United States of America
Satyajeet Nagargoje
  • Georgetown University, Washington, DC, United States of America
Chao Yan
  • Georgetown University, Washington, DC, United States of America

Funding

  • Golovnev, Alexander: Supported by the NSF CAREER award (grant CCF-2338730)
  • Hatami, Pooya: Supported by NSF grant CCF-1947546.
  • Nagargoje, Satyajeet: Supported by the NSF CAREER award (grant CCF-2338730)
  • Yan, Chao: Research supported in part by by a gift to Georgetown University.

Acknowledgements

We thank Omar Alrabiah, Jesse Goodman, Jonathan Mosheiff, and João Ribeiro for sharing with us an early draft of their work. We would also like to thank Jesse Goodman and S. Venkitesh for helpful discussions and pointers. We are very grateful to the anonymous reviewers for their comments and pointers to related work. Part of this work was conducted while the second author was visiting the Simons Institute for the Theory of Computing at UC Berkeley; he extends his thanks to the institute for its support and hospitality.

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