Polynomials, roots, and interlacing

TL;DR

This paper explores properties of polynomials with real and complex roots, focusing on transformations that preserve root structures, and extends these concepts to multivariable polynomials and stability regions.

Contribution

It introduces new insights into root-preserving transformations for polynomials in one and multiple variables, including stability considerations for complex roots.

Findings

Characterization of linear transformations preserving real-rootedness

Development of properties for multivariable polynomials with real roots

Analysis of stability regions for polynomials with complex roots

Abstract

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part covers polynomials in several variables that generalize polynomials with all real roots. We introduce generating functions and use them to establish properties of a linear transformation. We also consider matrices and matrix polynomials. The third part considers polynomials with complex roots. The two main classes considered are polynomials with all roots in the left half plane (stable polynomials) and those with all roots in the lower half plane (Upper half plane polynomials). These naturally generalize to polynomials in many variables. And, of course, there is much more.