A Standard Form in (some) Free Fields: How to construct Minimal Linear Representations

TL;DR

This paper introduces a standard form for elements in free fields that facilitates minimal linear representations, enabling advanced linear algebra techniques for non-commutative algebraic operations.

Contribution

It presents a new standard form for free field elements that simplifies constructing minimal linear representations for sums and products.

Findings

Enables use of linear algebra techniques in free fields

Provides a method for minimal linear representations of sums and products

Completes minimal arithmetic in free fields with inverse operations

Abstract

We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in standard form). This completes "minimal" arithmetics in free fields since "minimal" constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.