Polynomial-Time PIT from (Almost) Necessary Assumptions

Abstract

The celebrated result of Kabanets and Impagliazzo (Computational Complexity, 2004) showed that PIT algorithms imply circuit lower bounds, and vice versa. Since then it has been a major challenge to understand the precise connections between PIT and lower bounds. In particular, a main goal has been to understand which lower bounds suffice to obtain efficient PIT algorithms, and how close are they to lower bounds that are necessary for the conclusion. We construct polynomial-time PIT algorithms from lower bounds that are, up to relatively minor remaining gaps, necessary for the existence of such algorithms. That is, we prove that these lower bounds are, up to the mentioned minor gaps, both sufficient and necessary for polynomial-time PIT, over fields of characteristic zero. Over sufficiently large finite fields, we show a similar result wherein the PIT algorithm runs in time $n^{\log^{(c)}(n)}$, i.e. a power of $c$-iterated log for an arbitrarily large constant $c>1$. The key to these improvements is studying PIT versus lower bounds in the uniform setting, in which we focus on proving lower bounds for uniform arithmetic circuits and their variants (and on deducing algorithms from such lower bounds). Indeed, by working in this setting we obtain results that are significantly tighter than previously known results concerning polynomial-time PIT vs lower bounds, and are in fact also tighter than known hardness-vs-randomness connections in the Boolean setting. Our results are obtained by combining recent techniques from Boolean hardness vs randomness, and in particular the generator of Chen and Tell (FOCS 2021), with the algebraic hitting-set generator of Guo, Kumar, Saptharishi, and Solomon (SIAM J. Computing 2022) along with the bootstrapping ideas of Agrawal, Ghosh, and Saxena (STOC 2018) and of Kumar, Saptharishi, and Tengse (SODA 2019).