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

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 $\{P_d : d \in \mathbb{N}\}$ of polynomials in $\mathsf{VNP}$, where $P_d$ is of degree $d$ in $n = d^{O(1)}$ variables, such that over all finite fields $\mathbb{F}_q$, any homogeneous depth-$5$ circuit which computes $P_d$ must have size at least $\exp(Ω_q(\sqrt{d}))$. To the best of our knowledge, this is the first super-polynomial lower bound for this class for any field $\mathbb{F}_q \neq \mathbb{F}_2$. Our proof builds up on the ideas developed on the way to proving lower bounds for homogeneous depth-$4$ circuits [GKKS13, FLMS13, KLSS14, KS14] and for non-homogeneous depth-$3$ circuits over finite fields [GK98, GR00]. Our key insight is to look at the space of shifted partial derivatives of a polynomial as a space of functions from $\mathbb{F}_q^n \rightarrow \mathbb{F}_q$ 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 [KS14].