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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2007, Volume 19, Issue 6, Pages 184–199 (Mi aa152)

Research Papers

Dessins d'enfants and differential equations

b Department of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk

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.

Keywords: Riemann–Hilbert problem, Fuchsian equation, dessins d'enfants.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:6, 1003–1014

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MSC: 34M50
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Citation: F. Lárusson, T. Sadykov, “Dessins d'enfants and differential equations”, Algebra i Analiz, 19:6 (2007), 184–199; St. Petersburg Math. J., 19:6 (2008), 1003–1014

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Vitaly A. Krasikov, Timur M. Sadykov, “The Newton polytope of the optimal differential operator for an algebraic curve”, Zhurn. SFU. Ser. Matem. i fiz., 6:2 (2013), 200–210
2. Bishop Ch.J., “True Trees Are Dense”, Invent. Math., 197:2 (2014), 433–452
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