On the Factorization of Non-Commutative Polynomials (in Free Associative Algebras)

Abstract

We describe a simple approach to factorize non-commutative (nc) 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).

29 pages, extended (section 2.2 is new) and slightly updated version, accepted in JSC

References in corpus (4)

• Factorization theory: From commutative to noncommutative settings
• Factorizations of Elements in Noncommutative Rings: A Survey
• On Noncommutative Finite Factorization Domains
• A Factorization Algorithm for G-Algebras and Applications

Cited by in corpus (6)

• Geometry of free loci and factorization of noncommutative polynomials
• Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras
• Free Fractions: An Invitation to (applied) Free Fields
• A Factorization Theory for some Free Fields
• Horner Systems: How to efficiently evaluate non-commutative polynomials (by matrices)
• A Standard Form in (some) Free Fields: How to construct Minimal Linear Representations