Self-interlacing polynomials

TL;DR

This paper introduces self-interlacing polynomials, a new subclass of real polynomials with roots that mirror Hurwitz stability, and explores their properties, relations, and stability criteria.

Contribution

It defines self-interlacing polynomials, establishes their isomorphism with Hurwitz stable polynomials, and develops new stability criteria based on Hankel matrices.

Findings

Self-interlacing polynomials have roots that interlace with their conjugates.

A new criterion for Hurwitz stability using Hankel matrix minors is proposed.

Relations between self-interlacing polynomials, Hurwitz, and Hankel matrices are established.

Abstract

We describe a new subclass of the class of real polynomials with real simple roots called self-interlacing polynomials. This subclass is isomorphic to the class of real Hurwitz stable polynomials (all roots in the open left half-plane). In the work, we present basic properties of self-interlacing polynomials and their relations with Hurwitz and Hankel matrices as well as with Stiltjes type of continued fractions. We also establish "self-interlacing" analogues of the well-known Hurwitz and Li\'enard-Chipart criterions for stable polynomials. A criterion of Hurwitz stability of polynomials in terms of minors of certain Hankel matrices is established.