On Hitting-Set Generators for Polynomials that Vanish Rarely

The problem of constructing pseudorandom 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 pseudorandom generators, or even hitting-set generators, 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.