Dessins d’enfants and differential equations. (English. Russian original) Zbl 1206.14058
St. Petersbg. Math. J. 19, No. 6, 1003-1014 (2008); translation from Algebra Anal. 19, No. 6, 184-199 (2007).
Summary: 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.
MSC:
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
| 39A12 | Discrete version of topics in analysis |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 34M50 | Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain |
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