On the factorization of non-commutative polynomials (in free associative algebras). (September 2019)

• Record Type: Journal Article
• Authors: Schrempf, Konrad
• Is Part Of: Journal of symbolic computation. Volume 94(2019)
• Page Start: 126
• Page End: 148
• Publication Date: 2019-09
• Subjects: Free associative algebra -- Factorization of polynomials -- Minimal linear representation -- Companion matrix -- Free field -- Non-commutative formal power series; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2018.07.004
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: 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).