We discuss the relation between the linear Tschebyshev–Padé approximations to analytic function f and the diagonal type I Hermite–Padé polynomials for the tuple of functions [1, f1, f2] where the pair of functions f1, f2 forms certain Nikishin system. An approach is proposed of how to extend the seminal Stahl’s Theory for Padé approximations for multivalued analytic functions to the Tschebyshev–Padé approximations. The approach is based on the relation between Tschebyshev–Padé approximations and Hermite–Padé polynomials and also on a connection of Hermite– Padé polynomials and multipoint Padé approximants. Bibliography: [47] titles.

В работе рассматривается вопрос о распределении нулей полиномов Эрмита-Паде типа I, ассоциированных с вектор-функцией $\vec f=(f_1,…,f_s)$, компоненты $f_k$ которой являются функциями с конечным числом точек ветвления на плоскости. Предполагается, что множества ветвления компонент функции достаточно хорошо отделены друг от друга (случай Анжелеско). При этом условии доказывается теорема о предельном распределении нулей для таких полиномов. Предельные меры определяются в терминах стандартной векторной задачи равновесия. Доказательство теоремы основано на методах, разработанных Г. Шталем [59]-[63], A. A. Гончаром и автором настоящей работы [27], [55]. В настоящей работе указанные методы развиваются с целью приложения к наборам полиномов, определяемых системами комплексных соотношений ортогональности. Наряду с характеризацией предельных распределений нулей полиномов Эрмита-Паде, использующей векторную задачу равновесия, мы рассматриваем альтернативную характеризацию в терминах римановой поверхности $\mathcal R(\vec f )$, ассоциированной с $\vec f$. В этих терминах мы выдвигаем более общую (без условия Анжелеско) гипотезу о распределении нулей полиномов Эрмита-Паде. Библиография: 72 названия.

Теорема Гончара-Шталя о $\rho^2$ характеризует скорость сходимости наилучших (чебышeвских) равномерных рациональных приближений (со свободными полюсами) для одного из важнейших классов аналитических функций. Сама эта теорема, ее варианты и обобщения, методы, используемые в доказательстве, и прочие моменты составляют важную подобласть теории рациональных приближений аналитических функций и комплексного анализа. В статье вкратце очертены контуры этой подобласти. В центре изложения находится фундаментальный вклад А. А. Гончара и Г. Шталя в эту теорию. Библиография: 70 названий.

The Gonchar-Stahl rho(2)-theorem characterizes the rate of convergence of best uniform (Chebyshev) rational approximations (with free poles) for one basic class of analytic functions. The theorem itself, modifications and generalizations of it, methods involved in its proof and other related details constitute an important subfield in the theory of rational approximations of analytic functions and complex analysis.This paper briefly outlines the essentials of the subfield. The fundamental contributions of A. A. Gonchar and H. Stahl are at the heart of the exposition.

Type I Hermite--Pad\'e polynomials for a set of functions $f_0, f_1, ..., f_s$ at infinity, $Q_{n,0}$, $Q_{n,1}$, ..., $Q_{n,s}$, is defined by the asymptotic condition $$ R_n(z):=\bigl(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s\bigr)(z) =\mathcal O (\frac1{z^{s n+s}}), \quad z\to\infty, $$ with the degree of all $Q_{n,k}\leq n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\to \infty$ under the assumption that all $f_j$'s are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients. We discuss in more detail the case when $f_k$'s are powers of the same function $f$ ($f_k=f^k$); for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed. From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Pad\'e approximants (when $s=1$).

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.

The paper is devoted to a study of phase transitions in the Hermitian random matrix models with a polynomial potential. In an alternative equivalent language, we study families of equilibrium measures on the real line in a polynomial external field. The total mass of the measure is considered as the main parameter, which may be interpreted also either as temperature or time. Our main tools are differentiation formulas with respect to the parameters of the problem, and a representation of the equilibrium potential in terms of a hyperelliptic integral. Using this combination we introduce and investigate a dynamical system (system of ODE's) describing the evolution of families of equilibrium measures. On this basis we are able to systematically derive a number of new results on phase transitions, such as the local behavior of the system at all kinds of phase transitions, as well as to review a number of known ones.

УДК 517.53 Е. А. Рахманов, С. П. Суетин Распределение нулей полиномов Эрмита-Паде для пары функций, образующей систему Никишина В работе изучается распределение нулей полиномов Эрмита-Паде первого рода для пары функций с произвольным четным числом общих точек ветвления, расположенных на вещественной прямой в предположении, что эта пара функций образует обобщенную комплексную систему Никишина.Доказано (теорема 1), что предельное распределение нулей существует и совпадает с равновесной мерой компакта, обладающего S -свойством в гармоническом внешнем поле.Вопрос о существовании S -компакта решается в теореме 2.Основная идея доказательства теоремы 1 состоит в замене векторной теоретико-потенциальной задачи равновесия на скалярную задачу с внешним полем и последующем использовании общего метода Гончара-Рахманова, разработанного при решении "задачи об 1/9".Обсуждается связь полученных результатов с некоторыми результатами и гипотезами Наттолла.

E.A. Rakhmanov and S. P. Suetin 1. Let E = ⊔p j=1[α2j−1, α2j ] be a system of p disjoint closed intervals on the real axis R, where p > 1, and let wα(z) = √ (z − α1) · · · (z − α2p), so that wα(z) ∼ z as z →∞ in the domain D := C \E. For x ∈ E let w α (x) denote the limit value of wα(x+ iy) as y → +0. For a finite set Σβ = {β1, . . . , βq} ⊂ D with q > 2, let h : E → R be a holomorphic function on E (h ∈ H (E)) such that h extends as a multivalued analytic function from E along any path in the domain Ωβ := C \Σβ , but h / ∈ H (Ωβ). The class of such functions will be denoted by A ◦ E(Ωβ). Let K be an arbitrary compact subset of D which does not separate the complex plane and let D(K) = C \K. We shall assume that the domain D(K) is regular with respect to the solution of the Dirichlet problem. For a given function h ∈ A ◦ E(Ωβ) let Kh = Kh(E) denote the class of compact subsets K of D such that h extends from E to D(K) as a holomorphic (single-valued analytic) function.

We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials – polynomial solutions of second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest. The problem has a rich variety of connections with other fields of analysis; some of them are briefly mentioned in the paper.

This paper studies a variation of the equilibrium energy for a certain fairly general functional which appears naturally in the solution of many rational approximation problems of multi-valued analytic functions. The main result of this work states that for the energy functional under consideration and a certain class of admissible compact sets, related to the function to be approximated, the corresponding stationary compact set is fully characterized by the so-called S-property. Bibliography: 38 titles.