The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent

TL;DR

This paper explores the connection between polynomial identity testing and circuit lower bounds, proving a restricted case of the real τ-conjecture, which leads to lower bounds and deterministic testing algorithms for certain depth-4 circuits computing the permanent.

Contribution

It proves a restricted case of the real τ-conjecture, establishing lower bounds and a deterministic identity testing algorithm for a class of depth-4 circuits.

Findings

Restricted real τ-conjecture holds for certain sums of sparse polynomials.

Depth-4 circuits in this class cannot compute the permanent efficiently.

Deterministic polynomial identity testing algorithm developed for these circuits.

Abstract

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a "real {\tau}-conjecture" which is inspired by this connection. The real {\tau}-conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real {\tau}-conjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.