Efficient Identity Testing and Polynomial Factorization in Nonassociative Free Rings

In this paper we study arithmetic computations over non-associative, and non-commutative free polynomials ring F{x1, x2, . . . , xn}. Prior to this work, the non-associative arithmetic model of computation was considered by Hrubes, Wigderson, and Yehudayoff [HWY10]. They were interested in completeness and explicit lower bound results. We focus on two main problems in algebraic complexity theory: Polynomial Identity Testing (PIT) and polynomial factorization over F{x1, x2, . . . , xn}. We 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 non-commutative 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 the 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).