Dessins d’enfants and differential equations

Lárusson, F.; Sadykov, T.

doi:10.1090/S1061-0022-08-01033-9

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.

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