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 $mathbb{F}{x_1,x_2,ldots,x_n}$. 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 $mathbb{F}{x_1,x_2,ldots,x_n}$ and show the following results. (1) Given an arithmetic circuit $C$ of size $s$ computing a polynomial $fin mathbb{F} {x_1,x_2,ldots,x_n}$ 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 $fin mathbb{F} {x_1,x_2,ldots,x_n}$ 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 $mathbb{F}=mathbb{Q}$. Over finite fields of characteristic $p$, our algorithm runs in time $poly(n,s,d,p)$.