On the power of homogeneous depth 4 arithmetic circuits

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We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in VP. Our results hold for the Iterated Matrix Multiplication polynomial - in particular we show that any homogeneous depth 4 circuit computing the (1,1) entry in the product of n generic matrices of dimension nᵒ⁽¹⁾ must have size nΩ(√n). Our results strengthen previous works in two significant ways. Our lower bounds hold for a polynomial in VP. Prior to our work, Kayal et al [KLSS14] proved an exponential lower bound for homogeneous depth 4 circuits (over fields of characteristic zero) computing a poly in VNP. The best known lower bounds for a depth 4 homogeneous circuit computing a poly in VP was the bound of nΩ(n) by [LSS, KLSS14].Our exponential lower bounds also give the first exponential separation between general arithmetic circuits and homogeneous depth 4 arithmetic circuits. In particular they imply that the depth reduction results of Koiran [Koi12] and Tavenas [Tav13] are tight even for reductions to general homogeneous depth 4 circuits (without the restriction of bounded bottom fanin). Our lower bound holds over all fields. The lower bound of [KLSS14] worked only over fields of characteristic zero. Prior to our work, the best lower bound for homogeneous depth 4 circuits over fields of positive characteristic was nΩ(n) [LSS, KLSS14].