Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Computation of conformal representations of compact Riemann surfaces


Authors: Guillermo López Lagomasino, Domingo Pestana, José M. Rodríguez and Dmitry Yakubovich
Journal: Math. Comp. 79 (2010), 365-381
MSC (2000): Primary 30F99; Secondary 05E35, 30C30, 58C15
DOI: https://doi.org/10.1090/S0025-5718-09-02265-0
Published electronically: June 4, 2009
MathSciNet review: 2552231
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We find a system of two polynomial equations in two unknowns, whose solution allows us to give an explicit expression of the conformal representation of a simply connected three-sheeted compact Riemann surface onto the extended complex plane. This function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respect to so-called Nikishin systems of two measures.


References [Enhancements On Off] (What's this?)

  • 1. A.I. Aptekarev, G. López Lagomasino, and I.A. Rocha, Asymptotic behavier of the ratio of Hermite-Padé polynomials for Nikishin systems, Mat. Sb. 196 (2005), 1089-1107. MR 2188362 (2006h:41040)
  • 2. A.I. Aptekarev, V. Kalyagin, G. López Lagomasino, and I.A. Rocha, On the limit behavior of recurrence coefficients for multiple orthogonal polynomials, J. of Approx. Theory 139 (2006), 346-370. MR 2220045 (2007a:42048)
  • 3. S. A. Denisov, On Rakhmanov's theorem for Jacobi matrices, Proc. Amer. Math. Soc. 132 (2004), 847-852. MR 2019964 (2005h:47060)
  • 4. P. Deuflhard and A. Hofmann, Numerical analysis in modern scientific computing. An introduction (Second edition). Texts in Applied Mathematics, 43. Springer, New York, 2003. MR 1949263 (2003i:65001)
  • 5. A. López Garcıa and G. López Lagomasino, Ratio asymptotic of Hermite-Padé orthogonal polynomials for Nikishin systems. II, Advances in Math. 218 (2008), 1081-1106. MR 2419380
  • 6. P. Nevai, Orthogonal Polynomials, Mem. Amer. Math. Soc., no. 213, Providence, R. I., 1979. MR 519926 (80k:42025)
  • 7. J.M. Ortega and W. C. Rheinboldt, Iterative solutions of nonlinear equations in several variables, Acad. Press, N.Y.-London, 1970. MR 0273810 (42:8686)
  • 8. H. Poincaré, Les Méthodes Nouvelles de la Mécanique Celeste. Gauthier-Villars, Paris, 1892.
  • 9. E. A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the unit circle with weights not satisfying Szegő's condition, Math. USSR Sb. 58 (1987), 149-167. MR 0854969 (88b:42033)
  • 10. B. Shapiro and A. Vainstein, Counting real rational functions with all real critical values, Moscow Math. J. 3 no. 2 (2003), 647-659. MR 2025277 (2006d:26024)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 30F99, 05E35, 30C30, 58C15

Retrieve articles in all journals with MSC (2000): 30F99, 05E35, 30C30, 58C15


Additional Information

Guillermo López Lagomasino
Affiliation: Department of Mathematics, Universidad Carlos III de Madrid, 28911 Leganes, Spain
Email: lago@math.uc3m.es

Domingo Pestana
Affiliation: Department of Mathematics, Universidad Carlos III de Madrid, 28911 Leganes, Spain
Email: dompes@math.uc3m.es

José M. Rodríguez
Affiliation: Department of Mathematics, Universidad Carlos III de Madrid, 28911 Leganes, Spain
Email: jomaro@math.uc3m.es

Dmitry Yakubovich
Affiliation: Department of Mathematics, Universidad, Autónoma de Madrid and Instituto de Ciencias, Mathemáticas (CSIC-UAM-UC3M-UCM), Madrid, Spain
Email: dmitry.yakubovich@uam.es

DOI: https://doi.org/10.1090/S0025-5718-09-02265-0
Keywords: Orthogonal polynomials, compact Riemann surfaces, branched covering, nonlinear equations, Newtonian continuation method.
Received by editor(s): October 21, 2008
Received by editor(s) in revised form: February 13, 2009
Published electronically: June 4, 2009
Additional Notes: The first, second, and third authors’ research was partially supported by a grant from M.E.C. (MTM 2006-13000-C03-02) and a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain
The second and third authors’ research was partially supported by two grants from M.E.C. (MTM 2006-11976 and MTM 2007-30904-E), Spain
The fourth author’s research was partially supported by the Grant MTM2008-06621-C02-01, DGI-FEDER, of the Ministry of Science and Innovation, Spain
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society