On asymptotic Gauss-Lucas theorem

TL;DR

This paper extends the Gauss-Lucas theorem to sequences of polynomials with zeros mostly in a convex bounded domain, providing insights into the asymptotic behavior of their derivatives' zeros.

Contribution

It generalizes the classical Gauss-Lucas theorem to asymptotic cases involving sequences of polynomials with zeros confined to a convex domain.

Findings

Zeros of derivatives tend to cluster within the convex domain as polynomial degree increases

The extension applies to sequences where almost all zeros lie in the domain

Provides a framework for understanding zero distribution in asymptotic polynomial sequences

Abstract

In this note we extend the Gauss-Lucas theorem on the zeros of the derivative of a univariate polynomial to the case of sequences of univariate polynomials whose almost all zeros lie in a given convex bounded domain in C.