New Lower Bounds Against Homogeneous Non-Commutative Circuits

We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size ?(d/log d). For an n-variate polynomial with n > 1, the result can be improved to ?(nd), if d ? n, or ?(nd (log n)/(log d)), if d ? n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial