## 2006.320: Picard and Chazy solutions to the Painlevé VI equation

2006.320:
Marta Mazzocco
(2001)
*Picard and Chazy solutions to the Painlevé VI equation.*
Mathematische Annalen, 321 (1).
pp. 131-169.
ISSN 1432-1807

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DOI: 10.1007/PL00004500

## Abstract

We study the solutions of a particular family of Painlevé VI equations with parameters b = g = 0, d = \frac12 and 2a = (2m-1)2, for 2m Î \mathbb Z. We show that in the case of half-integer m, all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0,1,¥ and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVIm equation for any m such that 2m Î \mathbb Z. As an application, we classify all the algebraic solutions. For m half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For m integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions.

Item Type: | Article |
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Subjects: | MSC 2000 > 32 Several complex variables and analytic spaces MSC 2000 > 34 Ordinary differential equations MSC 2000 > 35 Partial differential equations MSC 2000 > 53 Differential geometry |

MIMS number: | 2006.320 |

Deposited By: | Miss Louise Stait |

Deposited On: | 16 August 2006 |

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