Geodesic
Dessins d'enfants and differential equations
Algebra i analiz, Tome 19 (2007) no. 6, pp. 184-199.

Voir la notice de l'article provenant de la source Math-Net.Ru

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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F. Lárusson; T. Sadykov. Dessins d'enfants and differential equations. Algebra i analiz, Tome 19 (2007) no. 6, pp. 184-199. http://geodesic.mathdoc.fr/item/AA_2007_19_6_a8/

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