Do Orthogonal Polynomials Dream of Symmetric Curves?

Abstract

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 others 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.

Supercritical regime for the kissing polynomials

“…The problem of existence of the S-contour, not only in the context of Gonchar and Rakhmanov's original work but also in other contexts, remained open until not so long ago, when Rakhmanov [38] outlined a very general max-min approach for obtaining S-contours. The rigorous analysis of this approach, nowadays called the Gonchar-Rakhmanov-Stahl program, depends heavily on the type of orthogonality weight and the geometry of the contours at hand, but nevertheless has been carried out in various different settings [5,28,31,33,44], many of which were largely inspired by Rakhmanov's outline in [38].…”

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

“…We see that this distinction between the one and two cut regimes will also play a fundamental role in the present analysis, as hinted at by Figure 2. This potential‐theoretic approach, known now as the Gonchar–Rakhmanov–Stahl (GRS) program, has been carried out in various scenarios, and we refer the reader to many excellent works on the subject 24,31–37 …”

“…This potential-theoretic approach, known now as the Gonchar-Rakhmanov-Stahl (GRS) program, has been carried out in various scenarios, and we refer the reader to many excellent works on the subject. 24,[31][32][33][34][35][36][37] Despite many successful applications of potential theory to the analysis of non-Hermitian orthogonal polynomials via the GRS program, we adopt an alternate viewpoint based on deformation techniques born from advances in the theory of random matrices and integrable systems. We will make heavy use of the technique known as continuation in parameter space, first developed in the context of integrable systems (cf.…”

Tschebyshev-Padé approximations for multivalued functions

“…In most cases such a compact exists and unique but there are exceptions and it is often not easy to establish fact of existence. In some cases existence may be proved by max-min energy problem (see [34], [30], [25], [36], [31], [32], [27]) which we discuss later in connection with our current problem (31). Anyway, situation is essentially simplified if existence of an S-compact set is known.…”