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Computation of highly ramified coverings

Author(s): Raimundas Vidunas; Alexander V. Kitaev.
Journal: Math. Comp. 78 (2009), 2371-2395.
MSC (2000): Primary 57M12, 34M55; Secondary 33E17
Posted: February 11, 2009
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Abstract: An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painlevé equation.

References:

1.
F. V. Andreev and A. V. Kitaev, Some examples of $ RS^2_3(3)$-transformations of ranks $ 5$ and $ 6$ as the higher order transformations for the hypergeometric function, Ramanujan J., 7, no. 4 (2003), pp. 455-476. MR 2040984 (2004k:33005)

2.
F. V. Andreev and A. V. Kitaev, Transformations $ {RS}_4^2(3)$ of the ranks $ \leq4$ and algebraic solutions of the sixth Painlevé equation, Comm. Math. Phys. 228 (2002), pp. 151-176. MR 1911252 (2003f:34186)

3.
E. A. Arnold, Modular algorithms for computing Gröbner bases, Journal of Symbolic Computation, 35, no. 4 (2003), pp. 403-419. MR 1976575 (2004c:13044)

4.
G. V. Belyi, Galois extensions of a maximal cyclotomic field (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 43, no. 2, (1979), pp. 267-276. English Translation in Math. USSR Izv. 14 (1980), pp. 247-256. MR 534593 (80f:12008)

5.
P. Boalch, The fifty-two icosahedral solutions to Painlevé. VI, J. Reine Angew. Math. 596 (2006), pp. 183-214. MR 2254812 (2007i:34149)

6.
P. Boalch, Some explicit solutions to the Riemann-Hilbert problem, in ``Differential Equations and Quantum Groups'', IRMA Lectures in Mathematics and Theoretical Physics, Vol. 9 (2006), pp. 85-112. MR 2322328 (2008h:34145)

7.
P. Boalch, Higher genus icosahedral Painlevé curves, Funk. Ekvac. 50 (2007), pp. 19-32. MR 2332077 (2008c:34188)

8.
J.-M. Couveignes, Tools for the computation of families of coverings, In ``Aspects of Galois theory'', London Math. Soc. Lecture Notes Ser., Vol. 256, Cambridge Univ. Press, 1999, pp. 38-65. MR 1708601 (2000f:14037)

9.
S. Diaz, R. Donagi, and D. Harbater, Every curve is a Hurwitz space, Duke Math. J., 59, no. 3 (1989), pp. 737-746. MR 1046746 (91i:14021)

10.
Ch. F. Doran, Algebraic and geometric isomonodromic deformations, J. Differential Geometry 59 (2001), pp. 33-85. MR 1909248 (2004d:32013)

11.
B. Dubrovin and M. Mazzocco, Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), pp. 55-147. MR 1767271 (2001j:34114)

12.
A. Grothendieck, Esquisse d'un programme, In Schneps L., Lochak P. (Eds.), ``Geometric Galois Actions I'', London Math. Soc. Lecture Note Ser., Vol. 242, Cambridge Univ. Press, 1997, pp. 5-48. English translation: the same volume, pp. 243-284. MR 1483107 (99c:14034)

13.
G.-M. Greuel, G. Pfister, H. Schönemann, SINGULAR 2.0.3. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2005). http://www.singular.uni-kl.de.

14.
R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)

15.
J. A. Hempel, Existence conditions for a class of modular subgroups of genus zero, Bull. Austr. Math. Soc. 66 (2002), pp. 517-525. MR 1939212 (2003j:20085)

16.
M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Physica 2D (1981), pp. 407-448. MR 625446 (83k:34010b)

17.
A. V. Kitaev, Special functions of isomonodromy type, rational transformations of the spectral parameter, and algebraic solutions of the Sixth Painlevé Equation (Russian), Algebra i Analiz 14, no. 3, (2002) pp. 121-139. English Translation in St. Petersburg Math. J. 14, no. 3, (2003) pp. 453-465. MR 1921990 (2003e:34168)

18.
A. V. Kitaev, Grothendieck's Dessins d'Enfants, Their Deformations and Algebraic Solutions of the Sixth Painlevé and Gauss Hypergeometric Equations, Algebra i Analiz 17, no. 1 (2005), pp. 224-273. MR 2140681 (2006b:33040)

19.
A. V. Kitaev, Remarks Towards Classification of $ RS_4^2(3)$-Transformations and Algebraic Solutions of the Sixth Painlevé Equation, Proceedings of the Angers Conference ``Asymptotic Theories and Painlevé Equations'' (June 01-05, 2004). Sèminaires et Congrès 14 (2006), pp. 199-227. See: smf.emath.fr/en/Publications/SeminairesCongres/2006/14/html.

MR 2353466

20.
E. Kreines, On families of geometric parasitic solutions for Belyi systems of genus zero, Fundamentalnaya i Priklandaya Matematika 9 (2003), pp. 103-111. Available at http://ellib.itep.ru/mathphys/psfiles/02__48.ps MR 2072622 (2005e:14038)

21.
A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovasz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), pp. 515-534. MR 682664 (84a:12002)

22.
K. Magaard, S. Shpectorov and H. Völklein, A GAP package for braid orbit computation and applications, Experimental Mathematics, 12 (2003), no. 4, pp. 385-393. MR 2043989 (2005e:12007)

23.
Mohamed El Marraki, Nicolas Hanusse, Jorg Zipperer, and Alexander Zvonkin, Cacti, braids and complex polynomials, Séminaire Lotharingien de Combinatoire 37 (1996). Available at http://citeseer.ist.psu.edu/259789.html. MR 1462334 (98j:57003)

24.
G. Shabat, On a class of families of Belyi functions, in ``Formal Power Series and Algebraic Combinatorics'', D. Krob, A. A. Mikhalev, A. V. Mikhalev (Eds.), Springer-Verlag, Berlin, Heidelberg, 2000; pp. 575-581. MR 1798251 (2001m:30007)

25.
L. Schneps, Dessins d'enfant on the Riemann sphere, In ``The Grothendieck theory of Dessins d'Enfant'', London Math. Soc. Lecture Notes Ser., Vol. 200, Cambridge Univ. Press, 1994; pp. 38-65. MR 1305393 (95j:11061)

26.
R. Vidūnas and A. V. Kitaev, Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions, Mathematische Nachrichten 280 (2007), pp. 1834-1855. MR 2365021.

27.
R. Vidūnas and A. V. Kitaev, Computation of $ RS$-pullback transformations for algebraic Painlevé VI solutions. Available at http://arxiv.org/abs/0705.2963.

28.
R. Vidūnas and A. V. Kitaev, Schlesinger transformations for algebraic Painlevé VI solutions. Available at http://arxiv.org/abs/0810.2766.

29.
R. Vidūnas, Algebraic Transformations of Gauss Hypergeometric Functions, Accepted by Funk. Ekvac. Available at http://www.arxiv.org/math.CA/0408269 (2004).

30.
R. Vidūnas, Transformations of some Gauss hypergeometric functions, J. Comp. Appl. Math.  178 (2005), pp. 473-487. MR 2127899 (2006a:33003)

31.
P. S. Wang, M. J. T. Guy and J. H. Davenport, $ P$-adic reconstruction of rational numbers, SIGSAM Bulletin, Vol. 16, ACM, 1982, pp. 2-3.

32.
L. Zapponi, Galois action on diameter four trees, preprint http://www.arxiv.org/math.AG/ 0108031 (2001).

33.
A. Zvonkin, Megamaps: Construction and Examples, Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, pp. 329-340. MR 1888783 (2003d:14036)

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Additional Information:

Raimundas Vidunas
Affiliation: Department of Mathematics, Kyushu University, Fukuoka 812-8581, Japan
Address at time of publication: Department of Mathematics, Kobe University, Rokko-dai 1-1, Nada-ku, Kobe 657-8501, Japan
Email: rvidunas@gmail.com

Alexander V. Kitaev
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Address at time of publication: Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
Email: kitaev@pdmi.ras.ru

DOI: 10.1090/S0025-5718-09-02233-9
PII: S 0025-5718(09)02233-9
Keywords: Belyi map, dessin d'enfant, the Painlev\'e VI equation.
Received by editor(s): June 21, 2007
Received by editor(s) in revised form: October 16, 2008
Posted: February 11, 2009
Additional Notes: The first author was supported by the 21st Century COE Programme ``Development of Dynamic Mathematics with High Functionality'' of the Ministry of Education, Culture, Sports, Science and Technology of Japan.
The second author was supported by JSPS grant-in-aide No. 14204012.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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