License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.7
URN: urn:nbn:de:0030-drops-126109
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12610/
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Doron, Dean ; Ta-Shma, Amnon ; Tell, Roei

On Hitting-Set Generators for Polynomials That Vanish Rarely

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LIPIcs-APPROX7.pdf (0.7 MB)


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.

BibTeX - Entry

@InProceedings{doron_et_al:LIPIcs:2020:12610,
  author =	{Dean Doron and Amnon Ta-Shma and Roei Tell},
  title =	{{On Hitting-Set Generators for Polynomials That Vanish Rarely}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Jaros{\l}aw Byrka and Raghu Meka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12610},
  URN =		{urn:nbn:de:0030-drops-126109},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.7},
  annote =	{Keywords: Hitting-set generators, Polynomials over finite fields, Quantified derandomization}
}

Keywords: Hitting-set generators, Polynomials over finite fields, Quantified derandomization
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)
Issue Date: 2020
Date of publication: 11.08.2020


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