Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter

Abstract

Let $C$ be a depth-3 circuit with $n$ variables, degree $d$, and top-fanin $k$ (called ${\Sigma\Pi\Sigma}(k,d,n)$ circuits) over base field ${\mathbb{F}}$. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests whether $C$ is identically zero. Klivans and Spielman [Proceedings of the 33rd Annual Symposium on Theory of Computing (STOC), 2001, pp. 216--223] observed that the problem is open even when $k$ is a constant. This case has been subjected to serious scrutiny over the past few years, starting from the work of Dvir and Shpilka [SIAM J. Comput., 36 (2007), pp. 1404--1434]. We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time ${\mbox{\rm poly}}(n)d^k$, regardless of the base field. The only field for which polynomial time algorithms were previously known is ${\mathbb{F}} = {\mathbb{Q}}$ [N. Kayal and S. Saraf, Proceedings of the 50th Annual Symposium on Foundations of Computer Science (FOCS), 2009, pp. 198--207; N. Saxena and C. Seshadhri, Proceedings of the 51st Annual Symposium on Foundations of Computer Science (FOCS), 2010, pp. 21--29]. This is the first blackbox algorithm for depth-3 circuits that does not use the rank-based approaches of Karnin and Shpilka [Proceedings of the 24th Annual Conference on Computational Complexity (CCC), 2009, pp. 274--285]. We prove an important tool for the study of depth-3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a ${\Sigma\Pi\Sigma}(k,d,n)$ circuit to $k$ variables but preserves the identity structure.