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

Konrad Schrempf

Research output: Contribution to journalArticle


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).
Original languageEnglish
Pages (from-to)1-22
Number of pages22
JournalarXiv.org e-Print archive
Publication statusPublished - 6 Jun 2017


  • math.RA
  • Primary 16K40, 16Z05, Secondary 16G99, 16S10


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