Homogeneous depth-five lower bounds for iterated matrix multiplication

Let \mathop{IMM}\nolimits _{n,n} be the (1,1) entry of a product of n independent n× n matrices of variables. Over every field of characteristic zero, every syntactically homogeneous \Sigma\Pi\Sigma\Pi\Sigma circuit computing \mathop{IMM}\nolimits _{n,n} has at least n^{\sqrt n/400} gates for all sufficiently large n, with an absolute threshold independent of the field. Bottom linear forms may have arbitrary support, and arbitrary finite fan-in, fan-out, and gate sharing are allowed. Over every field, a block expansion gives such circuits with at most n^{\sqrt n+4} gates for n ≥ 2. Thus the gate complexity over characteristic-zero fields is n^{Θ(\sqrt n)}.