On Algebraic Solutions to Painleve VI

TL;DR

This paper explores algebraic solutions to Painleve VI, introducing new conditions and the Tetrahedral Theorem, using moduli theory and dynamics on character varieties to classify solutions.

Contribution

It presents novel rationality and Diophantine conditions, and the Tetrahedral Theorem, advancing the classification of algebraic solutions to Painleve VI.

Findings

Rationality of parameters established

Trigonometric Diophantine conditions identified

Absence of solutions in certain cases proven

Abstract

We announce some results which might bring a new insight into the classification of algebraic solutions to the sixth Painleve equation. The main results consist of the rationality of parameters, trigonometric Diophantine conditions, and what the author calls the Tetrahedral Theorem regarding the absence of algebraic solutions in certain situations. The method is based on fruitful interactions between the moduli theoretical formulation of Painleve VI and dynamics on character varieties via the Riemann-Hilbert correspondence.