TY - JOUR

T1 - On the factorization of non-commutative polynomials (in free associative algebras)

AU - Schrempf, Konrad

N1 - Publisher Copyright:
© 2018 Elsevier Ltd
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - We describe a simple approach to factorize non-commutative polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear representations. To be more specific, a correspondence between factorizations of an element and upper right blocks of zeros in the system matrix (of its representation) is established. The problem is then reduced to solving a system of polynomial equations (with at most quadratic terms) with commuting unknowns to compute appropriate transformation matrices (if possible).

AB - We describe a simple approach to factorize non-commutative polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear representations. To be more specific, a correspondence between factorizations of an element and upper right blocks of zeros in the system matrix (of its representation) is established. The problem is then reduced to solving a system of polynomial equations (with at most quadratic terms) with commuting unknowns to compute appropriate transformation matrices (if possible).

KW - free associative algebra

KW - factorization of polynomials

KW - minimal linear representation

KW - companion matrix

KW - free field

KW - non-commutative formal power series

KW - free associative algebra

KW - factorization of polynomials

KW - minimal linear representation

KW - companion matrix

KW - free field

KW - non-commutative formal power series

KW - Free associative algebra

KW - Non-commutative formal power series

KW - Free field

KW - Companion matrix

KW - Factorization of polynomials

KW - Minimal linear representation

UR - http://www.scopus.com/inward/record.url?scp=85050863632&partnerID=8YFLogxK

U2 - 10.1016/j.jsc.2018.07.004

DO - 10.1016/j.jsc.2018.07.004

M3 - Article

SN - 0747-7171

VL - 94

SP - 126

EP - 148

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

ER -