A converse of the Gauss--Lucas theorem

TL;DR

This paper characterizes all linear operators on complex polynomials that reduce the zero set diameter, providing a converse perspective to the Gauss--Lucas theorem.

Contribution

It identifies and classifies all linear operators that decrease the zero set diameter of polynomials, offering a new understanding of zero distribution behavior.

Findings

Complete classification of diameter-decreasing linear operators

Extension of Gauss--Lucas theorem insights

New tools for analyzing polynomial zero sets

Abstract

All linear operators $L:\mathcal{C}[z]\to \mathcal{C}[z]$ which decrease the diameter of the zero set of any $P\in\mathcal{C}[z]$ are found.