On Algebraic Solutions to Painleve VI

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.

arXiv comment16 pages, 9 figures, 1 table. A contribution to the Proceedings of the Conference on Exact WKB Analysis and Microlocal Analysis in RIMS, Kyoto, May, 2008

References in corpus (4)

• Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I
• Transformations ${RS}_4^2(3)$ of the Ranks $\le 4$ and Algebraic Solutions of the Sixth Painlevé Equation
• Special Functions of the Isomonodromy Type, Rational Transformations of Spectral Parameter, and Algebraic Solutions of the Sixth Painlevé Equation
• Holomorphic dynamics, Painlevé VI equation and Character Varieties

Cited by in corpus (1)

• Flat Structure on the Space of Isomonodromic Deformations