Derandomizing the Isolation Lemma and Lower Bounds for Noncommutative Circuit Size

We give a randomized polynomial-time identity test for noncommutative circuits of polynomial degree based on the isolation lemma. Using this result, we show that derandomizing the isolation lemma implies noncommutative circuit size lower bounds. More precisely, we consider two restricted versions of the isolation lemma and show that derandomizing each of them implies nontrivial circuit size lower bounds for noncommutative circuits. These restricted versions of the isolation lemma are natural and would suffice for the standard applications of the isolation lemma.