On Matrix Multiplication and Polynomial Identity Testing

SIAM Journal on Computing, Ahead of Print.
Abstract. We show that lower bounds on the border rank of matrix multiplication can be used to nontrivially derandomize polynomial identity testing for small algebraic circuits. Letting [math] denote the border rank of [math] matrix multiplication, we construct a hitting set generator with seed length [math] that hits [math]-variate circuits of multiplicative complexity [math]. If the matrix multiplication exponent [math] is not 2, our generator has seed length [math] and hits circuits of size [math] for sufficiently small [math]. Surprisingly, the fact that [math] already yields new, nontrivial hitting set generators for circuits of sublinear multiplicative complexity.