Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter

TL;DR

This paper presents the first polynomial time blackbox identity testing algorithm for depth-3 circuits with bounded top fanin over any field, advancing the understanding of polynomial identity testing in algebraic complexity.

Contribution

It introduces a novel blackbox algorithm for depth-3 circuits with bounded top fanin that works over any field, independent of previous rank-based methods.

Findings

First polynomial time blackbox algorithm for depth-3 circuits with bounded top fanin.

Algorithm runs in time poly(nd^k), field-independent.

Develops a variable reduction transformation preserving circuit identities.

Abstract

Let C be a depth-3 circuit with n variables, degree d and top fanin k (called sps(k,d,n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005). We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(nd^k), regardless of the base field. The only field for which polynomial time algorithms were previously known is F=Q (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth-3 circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008). 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 sps(k,d,n) circuit to k variables, but preserves the identity structure.