An exponential lower bound for homogeneous depth-5 circuits over finite fields

Abstract

In this paper, we show exponential lower bounds for the class of homogeneous depth-5 circuits over all small finite fields. More formally, we show that there is an explicit family {Pd} of polynomials in VNP, where Pd is of degree d in n = dO(1) variables, such that over all finite fields GF(q), any homogeneous depth-5 circuit which computes Pd must have size at least [EQUATION].To the best of our knowledge, this is the first super-polynomial lower bound for this class for any non-binary field.Our proof builds up on the ideas developed on the way to proving lower bounds for homogeneous depth-4 circuits [Gupta et al., Fournier et al., Kayal et al., Kumar-Saraf] and for non-homogeneous depth-3 circuits over finite fields [Grigoriev-Karpinski, Grigoriev-Razborov]. Our key insight is to look at the space of shifted partial derivatives of a polynomial as a space of functions from GF(q)n to GF(q) as opposed to looking at them as a space of formal polynomials and builds over a tighter analysis of the lower bound of Kumar and Saraf [Kumar-Saraf].