An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits.

We prove a lower bound of $Omega(n^2/log^2 n)$ on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial $f(x_1, ldots, x_n)$. Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff [RSY08], who proved a lower bound of $Omega(n^{4/3}/log^2 n)$ for the same polynomial. Our improvement follows from an asymptotically optimal lower bound, in a certain range of parameters, for a generalized version of Galvinu0027s problem in extremal set theory.