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Algebra i Analiz, 2007, Volume 19, Issue 6, Pages 184–199 (Mi aa152)  

This article is cited in 2 scientific papers (total in 2 papers)

Research Papers

Dessins d'enfants and differential equations

F. Lárussona, T. Sadykovb

a School of Mathematical Sciences, University of Adelaide, Adelaide SA, Australia
b Department of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk

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.

Keywords: Riemann–Hilbert problem, Fuchsian equation, dessins d'enfants.

Full text: PDF file (247 kB)
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English version:
St. Petersburg Mathematical Journal, 2008, 19:6, 1003–1014

Bibliographic databases:

MSC: 34M50
Received: 31.10.2006
Language:

Citation: F. Lárusson, T. Sadykov, “Dessins d'enfants and differential equations”, Algebra i Analiz, 19:6 (2007), 184–199; St. Petersburg Math. J., 19:6 (2008), 1003–1014

Citation in format AMSBIB
\Bibitem{LarSad07}
\by F.~L\'arusson, T.~Sadykov
\paper Dessins d'enfants and differential equations
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 6
\pages 184--199
\mathnet{http://mi.mathnet.ru/aa152}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2411966}
\zmath{https://zbmath.org/?q=an:1206.14058}
\elib{https://elibrary.ru/item.asp?id=9942023}
\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 6
\pages 1003--1014
\crossref{https://doi.org/10.1090/S1061-0022-08-01033-9}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267497100009}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Vitaly A. Krasikov, Timur M. Sadykov, “The Newton polytope of the optimal differential operator for an algebraic curve”, Zhurn. SFU. Ser. Matem. i fiz., 6:2 (2013), 200–210  mathnet
    2. Bishop Ch.J., “True Trees Are Dense”, Invent. Math., 197:2 (2014), 433–452  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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