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

## TL;DR

This paper presents a straightforward method for factorizing non-commutative polynomials in free associative algebras by leveraging their minimal linear representations and solving related polynomial equations.

## Contribution

It introduces a novel approach linking factorizations to block structures in system matrices, simplifying the factorization process for non-commutative polynomials.

## Key findings

- Establishes a correspondence between factorizations and zero blocks in system matrices.
- Reduces factorization to solving polynomial equations with commuting unknowns.
- Provides a practical method for identifying irreducible factors in non-commutative polynomials.

## 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).