Efficient Identity Testing and Polynomial Factorization in Nonassociative Free Rings

Vikraman Arvind, Rajit Datta, Partha Mukhopadhyay, S. Raja, Marc Herbstritt
2017 International Symposium on Mathematical Foundations of Computer Science  
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 [7], 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
more » ... omputing a polynomial 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[13] 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).
doi:10.4230/lipics.mfcs.2017.38 dblp:conf/mfcs/ArvindDM017 fatcat:ehe7e6qbcrdljbtc6rltwaydaa