On Matrix Multiplication and Polynomial Identity Testing
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We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting R(n) denote the border rank of n × n × n matrix multiplication, we construct a hitting set generator with seed length O(√(n)·R^-1(s)) that hits n-variate circuits of multiplicative complexity s. If the matrix multiplication exponent ω is not 2, our generator has seed length O(n^1 - ε) and hits circuits of size O(n^1 + δ) for sufficiently small ε, δ > 0. Surprisingly, the fact that R(n) ≥ n^2 already yields new, non-trivial hitting set generators for circuits of sublinear multiplicative complexity.
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