In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring F x 1 x 2 x n . 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 x 1 x 2 x n and show the following results.

(1) Given an arithmetic circuit C of size s computing a polynomial f F x 1 x 2 x n of degree d , we give a deterministic pol y ( 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 x 1 x 2 x n of degree d , we give an efficient deterministic algorithm to compute circuits for the irreducible factors of f in time pol y ( n s d ) when F = Q . Over finite fields of characteristic p , our algorithm runs in time pol y ( n s d p ) .