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 $Ω(d/\log d)$. For an $n$-variate polynomial with $n>1$, the result can be improved to $Ω(nd)$, if $d\leq n$, or $Ω(nd \frac{\log n}{\log d})$, if $d\geq n$. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.

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