New Results on Noncommutative and Commutative Polynomial Identity Testing

Abstract

Using ideas from automata theory, we design the first polynomial deterministic identity testing algorithm for the sparse noncommutative polynomial identity testing problem. Given a noncommuting black-box polynomial $$f \in {\mathbb F}\{x_{1},\ldots,x_n\}$$of degree d with at most t monomials, where the variables xi are noncommuting, we give a deterministic polynomial identity test that checks if $$C \equiv 0$$and runs in time polynomial in d, n, |C|, and t. Our algorithm evaluates the black-box polynomial for xi assigned to matrices over $${\mathbb{F}}$$and, in fact, reconstructs the entire polynomial f in time polynomial in n, d and t. We apply this idea also to the reconstruction of black-box noncommuting algebraic branching programs (considered by Nisan (1995) and Raz and Shpilka (2005)) and obtain some results and connections to the problem of exact learning of noncommuting ABPs. Finally, we turn to commutative identity testing and explore the complexity of the problem when the coefficients of the input polynomial come from an arbitrary finite commutative ring with unity whose elements are uniformly encoded as strings and the ring operations are given by an oracle. We show that several algorithmic results for polynomial identity testing over fields also hold when the coefficients come from such finite rings.