On the limits of depth reduction at depth 3 over small finite fields

In a surprising recent result, Gupta-Kamath-Kayal-Saptharishi have proved that over Q any n(O(1)) -variate and n-degree polynomial in VPcan also be computed by a depth three Sigma Pi Sigma circuit of size 2(O)(root n log(3/2) n).(2) Over fixed-size finite fields, Grigoriev and Karpinski proved that any Sigma Pi Sigma circuit that computes the determinant (or the permanent) polynomial of a n xnmatrix must be of size 2(Omega(n)) . In this paper, for an explicit polynomial in VP(over fixed-size finite fields), we prove that any Sigma Pi Sigma circuit computing it must be of size 2(Omega(n log n)). The explicit polynomial that we consider is the iterated matrix multiplication polynomial of ngeneric matrices of size n x n. The importance of this result is that over fixed-size fields there is no depth reduction techniquethat can be used to compute all the n(O(1))-variate and n-degree polynomials in VPby depth 3 circuits of size 2(o(nlogn)). The result of Grigoriev and Karpinski can only rule out such a possibility for Sigma Pi Sigma circuits of size 2(o(n)). We also give an example of an explicit polynomial (NWn, is an element of(X)) in VNP (which is not known to be in VP), for which any Sigma Pi Sigma circuit computing it (over fixed-size fields) must be of size 2(Omega(nlogn)). The polynomial we consider is constructed from the combinatorial design of Nisan and Wigderson, and is closely related to the polynomials considered in many recent papers (by Kayal-Saha-Saptharishi, Kayal-Limaye-Saha-Srinivasan, and Kumar-Saraf), where strong depth 4 circuit size lower bounds are shown. (C) 2017 Elsevier Inc. All rights reserved.