On Algebraic Solutions to Painleve VI

Details On Algebraic Solutions to Painleve VI On Algebraic Solutions to Painleve VI

2008-09-09

arxiv.org/abs/0809.1482v3

Mathematics

Classical Analysis and ODEs

16 pages, 9 figures, 1 table. A contribution to the Proceedings of the Conference on Exact WKB Analysis and Microlocal Analysi

Scientific paper

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.

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