### 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).

Original language | English |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | arXiv.org e-Print archive |

Publication status | Published - 6 Jun 2017 |

### Keywords

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

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## Cite this

Schrempf, K. (2017). On the Factorization of Non-Commutative Polynomials (in Free Associative Algebras).

*arXiv.org e-Print archive*, 1-22.