On domains of noncommutative rational functions

TL;DR

This paper introduces the stable extended domain for noncommutative rational functions, showing it can be characterized by a monic linear pencil, and clarifies the relationship between domains and stable extended domains.

Contribution

It provides a complete description of the stable extended domain using minimal realizations, refining previous singularities theorems for noncommutative rational functions.

Findings

Stable extended domain characterized by monic linear pencil

Domains and stable extended domains coincide for functions regular at scalar points

Refinement of the singularities theorem by Kalyuzhnyi-Verbovetskyi and Vinnikov

Abstract

In this paper the stable extended domain of a noncommutative rational function is introduced and it is shown that it can be completely described by a monic linear pencil from the minimal realization of the function. This result amends the singularities theorem of Kalyuzhnyi-Verbovetskyi and Vinnikov. Furthermore, for noncommutative rational functions which are regular at a scalar point it is proved that their domains and stable extended domains coincide.