Poles of Painlev\'e IV Rationals and their Distribution

TL;DR

This paper investigates the distribution of poles and zeros of rational solutions to the Painlevé IV equation, linking them to roots of special polynomials and monodromy problems, and provides asymptotic analysis for large parameters.

Contribution

It connects the singularities of Painlevé IV rational solutions to roots of generalized Hermite and Okamoto polynomials via an inverse monodromy problem, offering new insights into their distribution.

Findings

Roots are described by an inverse monodromy problem for an anharmonic oscillator.

Roots are classified by monodromy representations of certain meromorphic functions.

Asymptotic distribution of roots is computed for large polynomial degrees.

Abstract

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlev\'e IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalised Hermite polynomials in the asymptotic regime when $m$ is large and $n$ fixed.