Finite branch solutions to Painleve VI around a fixed singular point

TL;DR

This paper classifies all finite branch solutions to Painleve VI near a fixed singularity, showing they are algebraic and providing a comprehensive classification up to Bäcklund transformations.

Contribution

It establishes that finite branch solutions are algebraic and classifies all such solutions up to Bäcklund transformations using algebraic geometry and related methods.

Findings

Finite branch solutions are algebraic solutions.

Global solutions are algebraic if and only if finitely many-valued.

Complete classification of finite branch solutions up to Bäcklund transformations.

Abstract

Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painleve VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Backlund transformations.