Near-optimal Bootstrapping of Hitting Sets for Algebraic Models
share
The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel [Ore22,DL78,Zip79,Sch80] 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 an explicit hitting set for all n-variate degree s, size s algebraic circuits of size (s+1)^n. In this paper, we prove the following results: - Let ϵ > 0 be a constant. For a sufficiently large constant n and all s > n, if we have an explicit hitting set of size (s+1)^n-ϵ for the class of n-variate degree s polynomials that are computable by algebraic circuits of size s, then for all s, we have an explicit hitting set of size s^∘ (O(^∗ s)) for s-variate circuits of degree s and size s. That is, if we can obtain a barely non-trivial exponent compared to the trivial (s+1)^n sized hitting set even for constant variate circuits, we can get an almost complete derandomization of PIT. - The above result holds when "circuits" are replaced by "formulas" or "algebraic branching programs". This extends a recent surprising result of Agrawal, Ghosh and Saxena [AGS18] who proved the same conclusion for the class of algebraic circuits, if the hypothesis provided a hitting set of size at most (s^n^0.5 - δ) (where δ>0 is any constant). Hence, our work significantly weakens the hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial saving over the trivial hitting set, and also presents the first such result for algebraic branching programs and formulas.
READ FULL TEXT