Dessins d'enfants and differential equations

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We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d'enfants. We show 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. We produce a universal annihilating operator for the inverses of a generic polynomial. We classify those plane trees that have a representation by Mobius transformations and 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 those plane trees whose Riemann-Hilbert problem has a hypergeometric solution of order at most two.
11 pages

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Complex Variables, Algebraic Geometry, Combinatorics, 32S40; 05C05, 14H30, 32G34, 34M15, 34M50

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http://salesiana.dossiersoluciones.com/handle/123456789/115204

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0607773
 http://arxiv.org/abs/math/0607773

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arXiv