Linear operators on polynomials preserving roots in open circular domains

TL;DR

This paper characterizes linear operators that preserve the location of polynomial roots within open circular domains, extending understanding of root-preserving transformations in complex analysis.

Contribution

It provides a complete description of operators preserving roots in open circular domains and clarifies differences between root-preserving classes for open versus closed sets.

Findings

Characterization of root-preserving operators in open circular domains

Comparison between operators preserving roots in open and closed sets

Answers to a question posed by Borcea and Branden

Abstract

In the present paper we answer a question raised by J. Borcea and P. Branden and give a description of the class of operators preserving roots in open circular domains, i.e., in images of the open upper half-plane under the Mobius transformations. Our second result is a description of the difference between $\Apr(G)$ (the class of operators preserving roots in an open set $G$) and $\Apr(\bar G)$ (the class of operators preserving roots in $\bar{G}$).