Abstract
The problem of constructing hittingset generators for polynomials of low degree is fundamental in complexity theory and has numerous wellknown applications. We study the following question, which is a relaxation of this problem: Is it easier to construct a hittingset 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 previouslyknown results. Our contributions are of two types:
 Over fields of size 2 ≤ 𝔽 ≤ poly(n), we show that the seed length of any hittingset 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 (nonexplicit) hittingset generator for polynomials of degree d ≤ n^{.99} that vanish on at most ε = 𝔽^{t} of their inputs with seed length O((dt)⋅log(n)). We also show a polynomialtime computable hittingset generator with seed length O((dt)⋅(2^{dt}+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 degreed closure is very large?", where the degreed closure of S is the variety induced by the set of degreed polynomials that vanish on S.
BibTeX  Entry
@InProceedings{doron_et_al:LIPIcs:2020:12610,
author = {Dean Doron and Amnon TaShma and Roei Tell},
title = {{On HittingSet Generators for Polynomials That Vanish Rarely}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
pages = {7:17:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771641},
ISSN = {18688969},
year = {2020},
volume = {176},
editor = {Jaros{\l}aw Byrka and Raghu Meka},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12610},
URN = {urn:nbn:de:0030drops126109},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.7},
annote = {Keywords: Hittingset generators, Polynomials over finite fields, Quantified derandomization}
}
Keywords: 

Hittingset 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 