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Algebra i Analiz, 2005, Volume 17, Issue 1, Pages 224–275 (Mi aa653)  

This article is cited in 20 scientific papers (total in 20 papers)

Research Papers

Grothendiecks dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations

A. V. Kitaevab

a Steklov Mathematical Institute, St. Petersburg, Russia
b School of Mathematics and Statistics, University of Sydney, Australia

Abstract: Grothendieck's dessins d'enfants are applied to the theory of the sixth Painlevé and Gauss hypergeometric functions, two classical special functions of isomonodromy type. It is shown that higher order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Belyĭ functions. Moreover, deformations of the dessins d'enfants are introduced, and it is shown that one-dimensional deformations are a useful tool for construction of algebraic sixth Painlevé functions.

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English version:
St. Petersburg Mathematical Journal, 2006, 17:1, 169–206

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Received: 25.09.2003

Citation: A. V. Kitaev, “Grothendiecks dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations”, Algebra i Analiz, 17:1 (2005), 224–275; St. Petersburg Math. J., 17:1 (2006), 169–206

Citation in format AMSBIB
\by A.~V.~Kitaev
\paper Grothendiecks dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlev\'e and Gauss hypergeometric equations
\jour Algebra i Analiz
\yr 2005
\vol 17
\issue 1
\pages 224--275
\jour St. Petersburg Math. J.
\yr 2006
\vol 17
\issue 1
\pages 169--206

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    This publication is cited in the following articles:
    1. A. V. Kitaev, “Quadratic transformations for the third and fifth Painlevé equations”, J. Math. Sci. (N. Y.), 136:1 (2006), 3586–3595  mathnet  crossref  mathscinet  zmath  elib  elib
    2. Ben Hamed B., Gavrilov L., “Families of Painlevé VI equations having a common solution”, Int. Math. Res. Not., 2005, no. 60, 3727–3752  mathscinet  zmath  isi
    3. Iwasaki K., Uehara T., “An ergodic study of Painlevé VI”, Math. Ann., 338:2 (2007), 295–345  crossref  mathscinet  zmath  isi  scopus
    4. Vidunas R., Kitaev A.V., “Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions”, Math. Nachr., 280:16 (2007), 1834–1855  crossref  mathscinet  zmath  isi  elib  scopus
    5. Kaneko K., Oyama Y., “Fifth Painlevé transcendents which are analytic at the origin”, Funkcial. Ekvac., 50:2 (2007), 187–212  crossref  mathscinet  zmath  isi  scopus
    6. Boalch Ph., “Some explicit solutions to the Riemann–Hilbert problem”, Differential Equations and Quantum Groups - ANDREY A. BOLIBRUKH MEMORIAL VOLUME, Irma Lectures in Mathematics and Theoretical Physics, 9, 2007, 85–112  mathscinet  zmath  isi
    7. Iwasaki K., “Finite branch solutions to Painlevé VI around a fixed singular point”, Adv. Math., 217:5 (2008), 1889–1934  crossref  mathscinet  zmath  isi  elib  scopus
    8. D. P. Novikov, “The $2{\times}2$ matrix Schlesinger system and the Belavin–Polyakov–Zamolodchikov system”, Theoret. and Math. Phys., 161:2 (2009), 1485–1496  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Vidūnas R., Kitaev A.V., “Computation of highly ramified coverings”, Math. Comp., 78:268 (2009), 2371–2395  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. Vidūnas R., “Algebraic Transformations of Gauss Hypergeometric Functions”, Funkcial. Ekvac., 52:2 (2009), 139–180  crossref  mathscinet  zmath  isi  scopus
    11. Kaneko K., “Local expansion of Painlevé VI transcendents around a fixed singularity”, J. Math. Phys., 50:1 (2009), 013531, 24 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. Movasati H., Reiter S., “Painlevé VI equations with algebraic solutions and family of curves”, Experiment. Math., 19:2 (2010), 161–173  crossref  mathscinet  zmath  isi  scopus
    13. Diarra K., “Construction and Classification of Certain Algebraic Solutions of the Garnier Systems”, Bull. Braz. Math. Soc., 44:1 (2013), 129–154  crossref  mathscinet  zmath  isi  elib  scopus
    14. Iorgov N., Lisovyy O., Tykhyy Yu., “Painlevé VI Connection Problem and Monodromy of C=1 Conformal Blocks”, J. High Energy Phys., 2013, no. 12, 029  crossref  mathscinet  zmath  isi  elib  scopus
    15. Vidunas R., Filipuk G., “A Classification of Coverings Yielding Heun-To-Hypergeometric Reductions”, Osaka J. Math., 51:4 (2014), 867–903  mathscinet  zmath  isi  elib
    16. Lisovyy O., Tykhyy Yu., “Algebraic Solutions of the Sixth Painlevé Equation”, J. Geom. Phys., 85 (2014), 124–163  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. J. Math. Sci. (N. Y.), 213:5 (2016), 706–722  mathnet  crossref  mathscinet
    18. Chen Zh., Kuo T.-J., Lin Ch.-Sh., Wang Ch.-L., “Green Function, Painlevé Vi Equation, and Eisenstein Series of Weight One”, J. Differ. Geom., 108:2 (2018), 185–241  crossref  mathscinet  zmath  isi
    19. Kato M., Mano T., Sekiguchi J., “Flat Structure and Potential Vector Fields Related With Algebraic Solutions to Painlevé Vi Equation”, Opusc. Math., 38:2 (2018), 201–252  crossref  mathscinet  isi  scopus
    20. van Hoeij M., Kunwar V.J., “Classifying (Almost)-Belyi Maps With Five Exceptional Points”, Indag. Math.-New Ser., 30:1 (2019), 136–156  crossref  mathscinet  zmath  isi  scopus
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