2006/07/30 by Finnur Larusson, Larusson, Finnur, Timur Sadykov +1
Mathematics · #05C05 #14H30 #32G34 #32S40 #34M15 #34M50 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CO #math.CV #msc:05C05 #msc:14H30 #msc:32G34 #msc:32S40 #msc:34M15 #msc:34M50
paper · pdf · doi:10.48550/arxiv.math/0607773
11 pages
arxiv created 2006/07/30 · arxiv updated 2009/12/01
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.