Painleve I, Coverings of the Sphere and Belyi Functions

TL;DR

This paper links Painleve I pole theory to the construction of special meromorphic functions with five ramification points, using limits of rational functions, including Belyi functions for the tritronquee solution.

Contribution

It introduces a novel approach to construct Painleve I solutions via limits of rational functions, connecting complex analysis, algebraic geometry, and special functions.

Findings

Constructed meromorphic functions with five ramification points as limits of rational functions.

Identified Belyi functions as rational approximations for the tritronquee solution.

Established a new link between Painleve I poles and algebraic functions.

Abstract

The theory of poles of solutions of Painleve-I is equivalent to the Nevanlinna problem of constructing a meromorphic function ramified over five points - counting multiplicities - and without critical points. We construct such meromorphic functions as limit of rational ones. In the case of the tritronquee solution these rational functions are Belyi functions.