×

Efficient computation of the branching structure of an algebraic curve. (English) Zbl 1238.14045

An efficient algorithm for computing the branching structure of an algebraic curve is given. To describe the branching structure one has to specify which sheets of the covering are connected in which way at a given branch point, i.e. one has to identify the monodromy.
Generators of the fundamental group of the base of the ramified covering punctured at the discriminant points of the curve are constructed via a minimal spanning tree of the discriminant points. This leads to paths of minimal length between the points. The branching structure is obtained by analytically continuing the roots of the equation defining the curve along the previously constructed generators of the fundamental group.

MSC:

14Q05 Computational aspects of algebraic curves
14H55 Riemann surfaces; Weierstrass points; gap sequences

Software:

MultRoot; Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. I. Bobenko, Introduction to Compact Riemann Surfaces, in: A. I. Bobenko and C. Klein (eds.), Computational Approach to Riemann Surfaces, Lect. Notes Math. 2013, 2011. · Zbl 1214.30030
[2] A. I. Bobenko and L. Bordag, Periodic multiphase solutions to the Kadomtsev-Petviashvili equation, J. Phys. A: Math. Gen. 22 (1989), 1259. · Zbl 0692.35082 · doi:10.1088/0305-4470/22/9/016
[3] E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994. · Zbl 0809.35001
[4] J.-M. Couveignes, Tools for the computation of families of coverings, in: Aspects of Galois Theory, Gainesville, FL, 1996, 38–65, London Math. Soc. Lecture Note Ser. 256, Cambridge Univ. Press, Cambridge, 1999.
[5] B. Deconinck, M. van Hoeij, Computing Riemann matrices of algebraic curves, Physica D, 152-153, (2001), 28–46. · Zbl 1054.14079 · doi:10.1016/S0167-2789(01)00156-7
[6] B. Deconinck and M. Patterson, Computing with plane algebraic curves, in: A. I. Bobenko and C. Klein (eds.), Computational Approach to Riemann Surfaces, Lect. Notes Math. 2013, 2011. · Zbl 1213.14114
[7] J. Frauendiener and C. Klein, Hyperelliptic theta-functions and spectral methods, J. Comp. Appl. Math. 167, (2004), 193. · Zbl 1052.65107 · doi:10.1016/j.cam.2003.10.003
[8] J. Frauendiener and C. Klein, Hyperelliptic theta-functions and spectral methods: KdV and KP solutions, Lett. Math. Phys. 76 (2006), 249–267. · Zbl 1127.14032 · doi:10.1007/s11005-006-0068-4
[9] –, Algebraic curves and Riemann Surfaces in Matlab, in: A. I. Bobenko and C. Klein (eds.), Computational Approach to Riemann Surfaces, Lect. Notes Math. 2013, 2011. · Zbl 1210.14070
[10] T. Grava and C. Klein, Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations, Comm. Pure Appl. Math. 60 (2007), 1623–1664. · Zbl 1139.65069 · doi:10.1002/cpa.20183
[11] C. Klein, A. Kokotov and D. Korotkin, Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces, Math. Zeitschr. 261(1) (2009), 73–108. · Zbl 1210.58025 · doi:10.1007/s00209-008-0314-9
[12] C. Klein and O. Richter, Ernst Equation and Riemann Surfaces, Lecture Notes in Physics 685, Springer, Berlin, 2005. · Zbl 1246.58014
[13] P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg de Vries equation, I,II,III, Comm. Pure Appl. Math. 36 (1983), 253–290, 571–593, 809–830. · Zbl 0532.35067 · doi:10.1002/cpa.3160360302
[14] J. R. Quine and P. Sarnak, P. (eds.), Extremal Riemann Surfaces, Contemporary Mathematics 201, AMS, 1997.
[15] M. Schmies, Computing Poincaré Theta series for Schottky groups, in: A. I. Bobenko and C. Klein (eds.), Computational Approach to Riemann Surfaces, Lect. Notes Math. 2013, 2011. · Zbl 1210.14003
[16] G. Springer, Introduction to Riemann Surfaces, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1957. · Zbl 0078.06602
[17] L. N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA, 2000. · Zbl 0953.68643
[18] C. L. Tretkoff and M. D. Tretkoff, Combinatorial group theory, Riemann surfaces and differential equations, Contemp. Math. 33 (1984), 467–517. · Zbl 0557.30036 · doi:10.1090/conm/033/767125
[19] www.comlab.ox.ac.uk/oucl/work/nick.trefethen
[20] Z. Zeng, Computing multiple roots of inexact polynomials, Math. Comp. 74 (2004), 869–903. · Zbl 1079.12007 · doi:10.1090/S0025-5718-04-01692-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.