Equivalence of Polynomial Identity Testing and Polynomial Factorization

In this paper, we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing. Specifically, we show that given an arithmetic circuit (either explicitly or via black-box access) that computes a multivariate polynomial f, the task of computing arithmetic circuits for the factors of f can be solved deterministically, given a deterministic algorithm for the polynomial identity testing problem (we require either a white-box or a black-box algorithm, depending on the representation of f). Together with the easy observation that deterministic factoring implies a deterministic algorithm for polynomial identity testing, this establishes an equivalence between these two central derandomization problems of arithmetic complexity. Previously, such an equivalence was known only for multilinear circuits (Shpilka & Volkovich, 2010).