Near-optimal Bootstrapping of Hitting Sets for Algebraic Circuits
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The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel states that any nonzero polynomial f(x_1,..., x_n) of degree at most s will evaluate to a nonzero value at some point on a grid S^n ⊆F^n with |S| > s. Thus, there is a deterministic polynomial identity test (PIT) for all degree-s 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 s^n^0.5 - δHuge(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 s^∘ (O( ^∗ s)). The authors asked if a similar conclusion followed if their hypothesis was weakened to having deterministic PIT with running time s^o(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 s^o(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 s^∘(O(^*s)) for all degree-s size-s algebraic circuits over n variables. In other words, any blackbox PIT with just a slightly non-trivial exponent of s compared to the trivial s^O(n) test can be used to give a nearly polynomial time blackbox PIT algorithm.
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