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

Konrad Schrempf

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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).
Original languageEnglish
Pages (from-to)126-148
Number of pages23
JournalJournal of Symbolic Computation
Publication statusPublished - 1 Sept 2019


  • free associative algebra
  • factorization of polynomials
  • minimal linear representation
  • companion matrix
  • free field
  • non-commutative formal power series
  • Free associative algebra
  • Non-commutative formal power series
  • Free field
  • Companion matrix
  • Factorization of polynomials
  • Minimal linear representation

Cite this