Finite branch solutions to Painleve VI around a fixed singular point

Details Finite branch solutions to Painleve VI around a fixed singular point Finite branch solutions to Painleve VI around a fixed singular point

2007-04-05

arxiv.org/abs/0704.0679v1

Mathematics

Algebraic Geometry

45 pages, 22 figures, 5 tables

Scientific paper

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.

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