Almost Cubic Bound for Depth Three Circuits in VP

In "An almost Cubic Lower Bound for Depth Three Arithmetic Circuits", [KST16] present an infinite family of polynomials in VNP, {Pn}n∈Z+ on n variables with degree n such that every ∑∏∑ circuit computing Pn is of size Ω̃(n3). A similar result was proven in [BLS16] for polynomials in VP with lower bound Ω ( n3 2 p logn ) . We present a modified polynomial and perform a tighter analysis to obtain an Ω̃(n3) lower bound for a family of polynomials in VP effectively bridging the VP and VNP results up to a log5 n factor. More generally, we show that for every N and D satisfying poly(N) > D > log2 N, there exist polynomials PN,D on N variable of degree D in VP that can not be computed by circuits of size Ω̃(N2D).