Root-counting Measures of Jacobi Polynomials and Topological Types and Critical Geodesics of Related Quadratic Differentials

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 (z 2 −1) 2 dz 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.

A refinement for rational functions of Pólya's method to construct Voronoi diagrams

“…Finally we note that our work (see Example 2) provides a family of counterexamples to Conjecture 1 in the recent paper by Shapiro and Solynin [19].…”

“…The characteristic polynomial of (5.5) is λ 2 − 2λ + 1 which has a double zero at λ = 1. Hence this example gives an instance of the necessity of the condition for uniform boundedness given in [4], and in addition a counterexample to Conjecture 1 of [19], which suggested that the condition in [4] would be unnecessary.…”

“…Example 2. The following incidentally gives a counterexample to Conjecture 1 of [19], described below. Let…”