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

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.