Near-optimal bootstrapping of hitting sets for algebraic circuits

Abstract

The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel states that any nonzero polynomial f(x1, . . ., xn) of degree at most s will evaluate to a nonzero value at some point on a grid Sn ⊆ Fn with |S| > s. Thus, there is a deterministic polynomial identity test (PIT) for all degrees size-s algebraic circuits in n variables that runs in time poly(s) • (s + 1)n. In a surprising recent result, Agrawal, Ghosh and Saxena (STOC 2018) showed any deterministic blackbox PIT algorithm for degree-s, size-s, n-variate circuits with running time as bad as sn0.5−δ Huge(n), where δ > 0 and Huge(n) is an arbitrary function, can be used to construct blackbox PIT algorithms for degree-s size s circuits with running time sexp(exp(O(log∗ s))). Agrawal et al. asked if a similar conclusion followed if their hypothesis was weakened to having deterministic PIT with running time so(n) • Huge(n). In this paper, we answer their question in the affirmative. We show that, given a deterministic blackbox PIT that runs in time so(n) • Huge(n) for all degree-s size-s algebraic circuits over n variables, we can obtain a deterministic blackbox PIT that runs in time sexp(exp(O(log∗ s))) for all degree-s size-s algebraic circuits over n variables. In other words, any blackbox PIT with just a slightly nontrivial exponent of s compared to the trivial sO(n) test can be used to give a nearly polynomial time blackbox PIT algorithm.