Efficient Identity Testing and Polynomial Factorization over Non-associative Free Rings

In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring F{x1, x2, . . . , xn}. Prior to this work, nonassociative arithmetic computation was considered by Hrubes, Wigderson, and Yehudayoff [HWY10], and they showed lower bounds and proved completeness results. We consider Polynomial Identity Testing (PIT) and polynomial factorization over F{x1, x2, . . . , xn} and show the following results. 1. Given an arithmetic circuit C of size s computing a polynomial f ∈ F{x1, x2, . . . , xn} of degree d, we give a deterministic poly(n, s, d) algorithm to decide if f is identically zero polynomial or not. Our result is obtained by a suitable adaptation of the PIT algorithm of Raz-Shpilka [RS05] for noncommutative ABPs. 2. Given an arithmetic circuit C of size s computing a polynomial f ∈ F{x1, x2, . . . , xn} of degree d, we give an efficient deterministic algorithm to compute circuits for the irreducible factors of f in time poly(n, s, d) when F = Q. Over finite fields of characteristic p, our algorithm runs in time poly(n, s, d, p).