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

V. Arvind, Rajit Datta, Partha Mukhopadhyay, S. Raja
2017 arXiv   pre-print
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
more » ... lynomial f∈F{x_1,x_2,...,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 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 poly(n,s,d) when F=Q. Over finite fields of characteristic p, our algorithm runs in time poly(n,s,d,p).
arXiv:1705.00140v2 fatcat:k7cermilhzawhn6qhcwlvbtwle