On hitting-set generators for polynomials that vanish rarely

Dean Doron, Amnon Ta-Shma, Roei Tell

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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) · (2dt+ 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.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020
EditorsJaroslaw Byrka, Raghu Meka
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771641
DOIs
StatePublished - 1 Aug 2020
Externally publishedYes
Event23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020 - Virtual, Online, United States
Duration: 17 Aug 202019 Aug 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume176
ISSN (Print)1868-8969

Conference

Conference23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020
Country/TerritoryUnited States
CityVirtual, Online
Period17/08/2019/08/20

Keywords

  • Hitting-set generators
  • Polynomials over finite fields
  • Quantified derandomization

ASJC Scopus subject areas

  • Software

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