On the distribution of zeros of the Hermite-Pade polynomials for three algebraic functions $1,f,f^2$ and the global topology of the Stokes lines for some differential equations of the third order

TL;DR

This paper heuristically explores the zero distribution of Hermite-Pade polynomials for specific algebraic functions, linking it to potential theory, differential equations, and the topology of Stokes lines, with implications for asymptotic analysis.

Contribution

It introduces a heuristic approach connecting zero distribution of Hermite-Pade polynomials to potential theory and differential equations, emphasizing the role of Stahl compact and Nuttall condenser structures.

Findings

Distribution of zeros relates to Chebotarev points and Stahl compact.

Connection established between zero distribution and Stokes line topology.

Heuristic method reduces complex problems to scalar potential problems.

Abstract

The paper presents some heuristic results about the distribution of zeros of Hermite-Pade polynomials of first kind for the case of three functions $1,f,f^2$, where $f$ has the form $f(z): = \prod\limits_ {j = 1 } ^3 (z-a_j) ^ {\alpha_j} $, $\alpha_j \in \mathbb C\setminus \mathbb Z $, $ \sum \limits_ {j = 1 } ^ 3 \alpha_j = 0 $, $ f (\infty) = 1 $. Answers are given in terms related to the problem of extreme minimum capacity of a plane Nuttall condenser, two plates of which intersect ("hooked" for each other) in a five points: the branch points $ a_1, a_2, a_3 $ of function $ f $, at $ v_1 = v $, where $ v = v (a_1, a_2, a_3) $ the Chebotarev point corresponding to triple points $ a_1, a_2, a_3 $, and another "unknown" at $ v_2 \neq v_1, a_1, a_2, a_3 $ (see Fig1-Fig3). The connection between the distribution of zeros of Hermite-Pade polynomials and global topology of the Stokes lines and the asymptotic behavior of Liouville-Green solutions of a class of homogeneous linear differential equations of the third order containing a large parameter is the free term is established. The basic idea of the new and still heuristic approach is to reduce at first some theoretical potential vector equilibrium problem to the scalar problem with the external field, and then use the general method of Gonchar-Rakhmanov developed in 1987 for solving the Varga problem "about $ 1/9 $". It is supposed that in the general case of an arbitrary algebraic function $ f $ it is imposible to constract the Nuttall condenser for a set of three functions $ 1, f, f ^ 2 $ without the knowledge of the structure of the Stahl compact for function $ f $. Namely, the Nuttall condenser is constructed using Green's function $ g_ {D} (z, \infty) $ for the Stahl domain $ D $ and "core" of Stahl compact, which consists of "effective" branch points of $ f $ and corresponding Chebotarev points.