Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin

Abstract

Shpilka & Wigderson (IEEE conference on computational complexity, vol 87, 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth-three arithmetic circuits with bounded bottom fanin over a field ${{\mathbb{F}}}$ of characteristic zero. We resolve this problem by proving a ${N^{\Omega(\frac{d}{\tau})}}$ lower bound for (nonhomogeneous) depth-three arithmetic circuits with bottom fanin at most ${\tau}$ computing an explicit ${N}$-variate polynomial of degree ${d}$ over ${{\mathbb{F}}}$. Meanwhile, Nisan & Wigderson (Comp Complex 6(3):217–234, 1997) had posed the problem of proving super-polynomial lower bounds for homogeneous depth-five arithmetic circuits. Over fields of characteristic zero, we show a lower bound of ${N^{\Omega(\sqrt{d})}}$ for homogeneous depth-five circuits (resp. also for depth-three circuits) with bottom fanin at most ${N^{\mu}}$, for any fixed ${\mu < 1}$. This resolves the problem posed by Nisan and Wigderson only partially because of the added restriction on the bottom fanin (a general homogeneous depth-five circuit has bottom fanin at most ${N}$).