Dessins d’enfants and differential equations
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- by F. Lárusson and T. Sadykov
- St. Petersburg Math. J. 19 (2008), 1003-1014
- DOI: https://doi.org/10.1090/S1061-0022-08-01033-9
- Published electronically: August 22, 2008
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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.References
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Bibliographic Information
- F. Lárusson
- Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
- MR Author ID: 347171
- 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
- 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 - © Copyright 2008 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 1003-1014
- MSC (2000): Primary 34M50
- DOI: https://doi.org/10.1090/S1061-0022-08-01033-9
- MathSciNet review: 2411966