## 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: F^{n} → 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}) · (_{2d}−_{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 ⊆ F^{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.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020 |

Editors | Jaroslaw Byrka, Raghu Meka |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771641 |

DOIs | |

State | Published - 1 Aug 2020 |

Externally published | Yes |

Event | 23rd 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 2020 → 19 Aug 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 176 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020 |
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Country/Territory | United States |

City | Virtual, Online |

Period | 17/08/20 → 19/08/20 |

## Keywords

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

## ASJC Scopus subject areas

- Software