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Dessins d'enfants and differential equations
Author(s):
F.
Lárusson;
T.
Sadykov
Original publication:
Algebra i Analiz,
tom 19
(2007),
nomer 6.
Journal:
St. Petersburg Math. J.
19
(2008),
1003-1014.
MSC (2000):
Primary 34M50
Posted:
August 22, 2008
MathSciNet review:
2411966
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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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Additional Information:
F.
Lárusson
Affiliation:
School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
Email:
finnur.larusson@adelaide.edu.au
T.
Sadykov
Affiliation:
Department of Mathematics and Computer Science, Siberian Federal University, Svobodnyj Prospekt 79, Krasnoyarsk 660041, Russia
Email:
sadykov@lan.krasu.ru
DOI:
10.1090/S1061-0022-08-01033-9
PII:
S 1061-0022(08)01033-9
Keywords:
Riemann--Hilbert problem,
Fuchsian equation,
dessins d'enfants.
Received by editor(s):
31/OCT/2006
Posted:
August 22, 2008
Additional Notes:
The first author was supported in part by the Natural Sciences and Engineering Research Council of Canada.
The second author was supported in part by the RFBR (grant no.~05-01-00517), by grant MK-851.2006.1 of the President of the Russian Federation, and by scientific educational grant no.~45.2007 from the Siberian Federal University. Part of the work of both authors was done at the University of Western Ontario
Copyright of article:
Copyright
2008,
American Mathematical Society
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