Algebraic equation and quadratic differential related to generalized Bessel polynomials with varying parameters

TL;DR

This paper investigates the zero distribution of generalized Bessel polynomials with varying parameters, linking it to quadratic differentials and their critical trajectories in the complex plane.

Contribution

It provides a necessary and sufficient condition for the existence of short trajectories of quadratic differentials associated with these polynomials.

Findings

Zeros cluster on a curve in the complex plane

Characterization of critical trajectories of quadratic differentials

Necessary and sufficient conditions for short trajectories

Abstract

The limiting set of zeros of generalized Bessel polynomials with varying parameters depending on the degree n cluster in a curve on the complex plane, which is a finite critical trajectory of a quadratic differential in the form {\lambda}^2(((z-a)(z-b))/(z^4))dz^2. The motivation of this paper is the description of the critical graphs of these quadratic differentials. In particular, we give a necessary and sufficient condition on the existence of short trajectories.