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

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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.

**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)**

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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.