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

 

Dessins d'enfants and differential equations


Authors: F. Lárusson and T. Sadykov
Original publication: Algebra i Analiz, tom 19 (2007), nomer 6.
Journal: St. Petersburg Math. J. 19 (2008), 1003-1014
MSC (2000): Primary 34M50
Published electronically: August 22, 2008
MathSciNet review: 2411966
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Abstract: A discrete version of the classical Riemann-Hilbert problem is stated and solved. In particular, a Riemann-Hilbert problem is associated with every dessin d'enfants. It is shown how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial. A universal annihilating operator for the inverses of a generic polynomial is produced. A classification is given for the plane trees that have a representation by Möbius transformations and for those that have a linear representation of dimension at most two. This yields an analogue for trees of Schwarz's classical list, that is, a list of the plane trees whose Riemann-Hilbert problem has a hypergeometric solution of order at most two.


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Additional Information

F. Lárusson
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
Email: finnur.larusson@adelaide.edu.au

T. Sadykov
Affiliation: Department of Mathematics and Computer Science, Siberian Federal University, Svobodnyj Prospekt 79, Krasnoyarsk 660041, Russia
Email: sadykov@lan.krasu.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-08-01033-9
PII: S 1061-0022(08)01033-9
Keywords: Riemann--Hilbert problem, Fuchsian equation, dessins d'enfants.
Received by editor(s): October 31, 2006
Published electronically: August 22, 2008
Additional Notes: The first author was supported in part by the Natural Sciences and Engineering Research Council of Canada.
The second author was supported in part by the RFBR (grant no. 05-01-00517), by grant MK-851.2006.1 of the President of the Russian Federation, and by scientific educational grant no. 45.2007 from the Siberian Federal University. Part of the work of both authors was done at the University of Western Ontario
Article copyright: © Copyright 2008 American Mathematical Society