On hitting-set generators for polynomials that vanish rarely

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}) · (_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 ⊆ 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.