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### Efficient Identity Testing and Polynomial Factorization in Nonassociative Free Rings

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### Abstract

In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring F{X}. Prior to this work, nonassociative arithmetic computation was considered by Hrubes, Wigderson, and Yehudayoff, and they showed lower bounds and proved completeness results. We consider Polynomial Identity Testing and Polynomial Factorization in F{X} and show the following results. 1. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give a deterministic polynomial algorithm to decide if f is identically zero. Our result is obtained by a suitable adaptation of the PIT algorithm of Raz and Shpilka for noncommutative ABPs. 2. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give an efficient deterministic algorithm to compute circuits for the irreducible factors of f in polynomial time when F is the field of rationals. Over finite fields of characteristic p, our algorithm runs in time polynomial in input size and p.

### BibTeX - Entry

```@InProceedings{arvind_et_al:LIPIcs:2017:8069,
author =	{Vikraman Arvind and Rajit Datta and Partha Mukhopadhyay and S. Raja},
title =	{{Efficient Identity Testing and Polynomial Factorization in  Nonassociative Free Rings}},
booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
pages =	{38:1--38:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-046-0},
ISSN =	{1868-8969},
year =	{2017},
volume =	{83},
editor =	{Kim G. Larsen and Hans L. Bodlaender and Jean-Francois Raskin},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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