Bootstrapping variables in algebraic circuits

Abstract

We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One only need to consider size-s degree-s circuits that depend on the first log∘ c s variables (where c is a constant and we are composing c logarithms). Thus, hitting-set generator (hsg) manifests a bootstrapping behavior— a partial hsg against very few variables can be efficiently grown to a complete hsg. A boolean analog, or a pseudorandom generator property of this type, is unheard-of. Our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially wrt variables. This is repeated c times in an efficient way. Pushing the envelope further we show that: (1) a quadratic-time blackbox PIT for 6913-variate degree-s size-s polynomials, will lead to a “near”-complete derandomization of PIT, and (2) a blackbox PIT for n-variate degree-s size-s circuits in snδ-time, for δ<1/2, will lead to a “near”-complete derandomization of PIT (in contrast, sn-time is trivial). Our second idea is to study depth-4 circuits that depend on constantly many variables. We show that a polynomial-time computable, O(s1.49)-degree hsg for trivariate depth-4 circuits bootstraps to a quasipolynomial time hsg for general poly-degree circuits, and implies a lower bound that is a bit stronger than Kabanets-Impagliazzo (STOC 2003).