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On Hitting-Set Generators for Polynomials That Vanish Rarely

Authors Dean Doron, Amnon Ta-Shma, Roei Tell

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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Hitting-set generators
  • Polynomials over finite fields
  • Quantified derandomization

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Abstract

The problem of constructing hitting-set generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct a hitting-set generator for polynomials p: 𝔽ⁿ → 𝔽 of degree d if we are guaranteed that the polynomial vanishes on at most an ε > 0 fraction of its inputs? We will specifically be interested in tiny values of ε≪ d/|𝔽|. This question was first considered by Goldreich and Wigderson (STOC 2014), who studied a specific setting geared for a particular application, and another specific setting was later studied by the third author (CCC 2017). 
In this work our main interest is a systematic study of the relaxed problem, in its general form, and we prove results that significantly improve and extend the two previously-known results. Our contributions are of two types:  
- Over fields of size 2 ≤ |𝔽| ≤ poly(n), we show that the seed length of any hitting-set generator for polynomials of degree d ≤ n^{.49} that vanish on at most ε = |𝔽|^{-t} of their inputs is at least Ω((d/t)⋅log(n)). 
- Over 𝔽₂, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d ≤ n^{.99} that vanish on at most ε = |𝔽|^{-t} of their inputs with seed length O((d-t)⋅log(n)). We also show a polynomial-time computable hitting-set generator with seed length O((d-t)⋅(2^{d-t}+log(n))). 
In addition, we prove that the problem we study is closely related to the following question: "Does there exist a small set S ⊆ 𝔽ⁿ whose degree-d closure is very large?", where the degree-d closure of S is the variety induced by the set of degree-d polynomials that vanish on S.

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Dean Doron, Amnon Ta-Shma, and Roei Tell. On Hitting-Set Generators for Polynomials That Vanish Rarely. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.7

Author Details

Dean Doron
  • Department of Computer Science, Stanford University, CA, USA
Amnon Ta-Shma
  • The Blavatnik School of Computer Science, Tel-Aviv University, Israel
Roei Tell
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel

Funding

  • Doron, Dean: Supported by a Motwani Postdoctoral Fellowship and by NSF award CCF-1763311.
  • Ta-Shma, Amnon: Supported by ISF grant 18/952 and by Len Blavatnik and the Blavatnik Family foundation.
  • Tell, Roei: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819702).

Acknowledgements

We are grateful to an exceptionally helpful anonymous reviewer, who pointed us to the work of Nie and Wang [Nie and Wang, 2015] and to follow-up works, suggested an alternative proof strategy for the lower bound in the special case of prime fields (the alternative proof for this special case appears in the full version), and helped improve our initial results.

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