Stable noncommutative polynomials and their determinantal representations

TL;DR

This paper characterizes stable noncommutative polynomials through determinantal representations involving strongly stable linear matrix pencils, linking stability to matrix pencil structures.

Contribution

It provides a novel characterization of stable noncommutative polynomials using strongly stable pencils and extends structure certificates to rational functions.

Findings

Stable polynomials correspond to determinantal representations with strongly stable pencils

Characterization of stability via matrix pencil structures

Extension of certificates to noncommutative rational functions

Abstract

A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of strongly stable linear matrix pencils, i.e., pencils of the form $H+iP_0+P_1x_1+\cdots+P_dx_d$, where $H$ is hermitian and $P_j$ are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a strongly stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix pencils and noncommutative rational functions.