Near-Optimal Bootstrapping of Hitting Sets for Algebraic Models

TL;DR

This paper demonstrates that even a slight improvement in explicit hitting set size for algebraic circuits can lead to near-complete derandomization of polynomial identity testing, extending results to formulas and algebraic branching programs.

Contribution

It weakens previous hypotheses needed for derandomization of PIT, showing minimal improvements can yield almost complete derandomization across various algebraic models.

Findings

A barely non-trivial hitting set size leads to near-complete derandomization.

Results extend from algebraic circuits to formulas and algebraic branching programs.

Weakens prior hypotheses, requiring only slight improvements for significant derandomization.

Abstract

$ \newcommand{\inparen}[1]{\left( #1 \right)} \newcommand{\pfrac}[2]{\inparen{\frac{1}{2}}} \newcommand{\ilog}[1]{\log^{\circ #1}} \newcommand{\F}{\mathbb{F}} $The Polynomial Identity Lemma (also called the "Schwartz--Zippel lemma") states that any nonzero polynomial $f(x_1,\ldots, x_n)$ of degree at most $s$ will evaluate to a nonzero value at some point on any grid $S^n \subseteq \F^n$ 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: $\bullet$ Let $\epsilon > 0$ be a constant. For a sufficiently large constant $n$, and all $s > n$, if we have an explicit hitting set of size $(s+1)^{n-\epsilon}$ for the class of $n$-variate degree-$s$ polynomials that are computable by algebraic circuits of size $s$, then for all large $s$, we have an explicit hitting set of size $s^{\exp(\exp (O(\log^\ast s)))}$ for $s$-variate circuits of degree $s$ and size $s$. That is, if we can obtain a barely non-trivial exponent (a factor-$s^{\Omega(1)} $ improvement) compared to the trivial $(s+1)^{n}$-size hitting set even for constant-variate circuits, we can get an almost complete derandomization of PIT. $\bullet$ 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 (STOC 2018, PNAS 2019) who proved the same conclusion for the class of algebraic circuits, if the hypothesis provided a hitting set of size at most $\inparen{s^{n^{0.5 - \delta}}}$ (where $\delta> 0$ 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 formulas.