Do orthogonal polynomials dream of symmetric curves?

TL;DR

This paper explores the zero distributions of complex orthogonal polynomials, revealing their tendency to align on specific curves, and discusses both established results and open questions in the theory of non-hermitian orthogonal polynomials.

Contribution

It provides new numerical insights into zero configurations, discusses rigorous results and conjectures, and highlights open problems in the limit zero distribution theory.

Findings

Zeros tend to align on certain curves.

Some configurations are rigorously explained.

Other behaviors remain conjectural or empirically observed.

Abstract

The complex or non-hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion. The problem of the limit zero distribution for the non-hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation. Thus, the second motivation of this paper is to discuss some "mysterious" configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while other are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.