Root-counting measures of Jacobi polynomials and topological types and critical geodesics of related quadratic differentials

TL;DR

This paper investigates the asymptotic zero distributions of Jacobi polynomials and classifies the topological types of their associated quadratic differential trajectories, revealing new insights into their geometric and analytical properties.

Contribution

It provides new results on the limit measures of Jacobi polynomial zeros and a complete classification of critical geodesic topologies for related quadratic differentials.

Findings

Support of the limit measure lies on critical trajectories of a specific quadratic differential.

Complete classification of topological types of critical geodesics based on complex parameters.

New insights into the asymptotic behavior and geometric structure of Jacobi polynomial zeros.

Abstract

Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form Q(z)dz^2=(az^2+bz+c)dz^2/(z^2-1)^2. Then we will give a complete classification, in terms of complex parameters a, b, and c, of possible topological types of critical geodesics for the quadratic differential of this type.