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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: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12610/
Doron, Dean ;
Ta-Shma, Amnon ;
Tell, Roei
On Hitting-Set Generators for Polynomials That Vanish Rarely
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: ?ⁿ → ? 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/|?|. 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 ≤ |?| ≤ 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 = |?|^{-t} of their inputs is at least Ω((d/t)⋅log(n)).
- Over ?₂, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d ≤ n^{.99} that vanish on at most ε = |?|^{-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 ⊆ ?ⁿ 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}
}
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Keywords: |
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Hitting-set generators, Polynomials over finite fields, Quantified derandomization |
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Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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11.08.2020 |
