Dessins d'enfants and differential equations

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dc.creatorLarusson, Finnur
dc.creatorSadykov, Timur
dc.date2006-07-30
dc.date.accessioned2026-07-07T07:21:09Z
dc.date.available2026-07-07T07:21:09Z
dc.descriptionWe 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.
dc.description11 pages
dc.identifierhttps://arxiv.org/abs/math/0607773
dc.identifierhttp://arxiv.org/abs/math/0607773
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/115204
dc.subjectComplex Variables
dc.subjectAlgebraic Geometry
dc.subjectCombinatorics
dc.subject32S40; 05C05, 14H30, 32G34, 34M15, 34M50
dc.titleDessins d'enfants and differential equations
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