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: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 𝔽_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=|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⊆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.