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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Grothendieck's dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations


Author: A. V. Kitaev
Original publication: Algebra i Analiz, tom 17 (2005), nomer 1.
Journal: St. Petersburg Math. J. 17 (2006), 169-206
MSC (2000): Primary 34M55, 33E17, 33E30
Published electronically: January 23, 2006
MathSciNet review: 2140681
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Abstract: Grothendieck's dessins d'enfants are applied to the theory of the sixth Painlevé and Gauss hypergeometric functions, two classical special functions of iso- monodromy type. It is shown that higher-order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Belyi functions. Moreover, deformations of the dessins d'enfants are introduced, and it is shown that one-dimensional deformations are a useful tool for construction of algebraic sixth Painlevé functions.


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Additional Information

A. V. Kitaev
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia, and School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
Email: kitaev@pdmi.ras.ru, kitaev@maths.usyd.edu.au

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00899-5
PII: S 1061-0022(06)00899-5
Keywords: Algebraic function, dessin d'enfant, hypergeometric function, isomonodromy deformation, Schlesinger deformations, the sixth Painlev\'e equation
Received by editor(s): September 25, 2003
Published electronically: January 23, 2006
Dedicated: Dedicated to Ludwig Dmitrievich Faddeev on the occasion of his 70th birthday
Article copyright: © Copyright 2006 American Mathematical Society