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Computation of the topological type of a real Riemann surface. (English) Zbl 1298.14067

The authors present an algorithm for the computation of the topological type of a real Riemann surface with an antiholomorphic involution \(\tau\) associated to an algebraic curve.
The topological type comprises genus, number of connected components of the set of fixpoints of \(\tau\), and whether or not these connected components divide the Riemann surface into one or two connected components.
An algorithm by C. L. Tretkoff and M. D. Tretkoff [in: Contributions to group theory, Contemp. Math. 33, 467–519 (1984; Zbl 0557.30036)] is used to compute a canonical homology basis and the genus of the surface. The topology type is determined by transforming the canonical homology basis into a homology basis whose \(\mathcal{A}\)-cycles are invariant under \(\tau\).
The authors demonstrate how a Matlab implementation of the algorithm can be applied to various examples of algebraic curves.

MSC:

14Q05 Computational aspects of algebraic curves
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 0557.30036
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[1] V. I. Arnol\(^{\prime}\)d, The situation of ovals of real plane algebraic curves, the involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms, Funkcional. Anal. i Priložen. 5 (1971), no. 3, 1 – 9 (Russian).
[2] Dennis S. Arnon and Scott McCallum, A polynomial-time algorithm for the topological type of a real algebraic curve, J. Symbolic Comput. 5 (1988), no. 1-2, 213 – 236. · Zbl 0664.14017
[3] E. Belokolos, A. Bobenko, V. Enolskii, A. Its, V. Matveev, Algebro-geometric approach to nonlinear integrable equations, Springer Series in nonlinear dynamics (1994). · Zbl 0809.35001
[4] Alexander I. Bobenko and Christian Klein , Computational approach to Riemann surfaces, Lecture Notes in Mathematics, vol. 2013, Springer, Heidelberg, 2011. · Zbl 1207.14002
[5] H. W. Braden and T. P. Northover, Klein’s curve, J. Phys. A 43 (2010), no. 43, 434009, 17. · Zbl 1200.14064
[6] H. Braden, V. Enolskii, T. Northover, Maple packages extcurves and CyclePainter, available at gitorious.org.
[7] E. Bujalance, A. F. Costa, and D. Singerman, Application of Hoare’s theorem to symmetries of Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 307 – 322. · Zbl 0794.30029
[8] Emilio Bujalance, Francisco Javier Cirre, José Manuel Gamboa, and Grzegorz Gromadzki, Symmetries of compact Riemann surfaces, Lecture Notes in Mathematics, vol. 2007, Springer-Verlag, Berlin, 2010. · Zbl 1208.30002
[9] S. A. Broughton, E. Bujalance, A. F. Costa, J. M. Gamboa, and G. Gromadzki, Symmetries of Riemann surfaces on which \?\?\?(2,\?) acts as a Hurwitz automorphism group, J. Pure Appl. Algebra 106 (1996), no. 2, 113 – 126. · Zbl 0847.30026
[10] Helena B. Campos, Basis of homology adapted to the trigonal automorphism of a Riemann surface, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 101 (2007), no. 2, 167 – 172 (English, with Spanish summary). · Zbl 1149.30032
[11] A. Comessati, Sulla connessione delle superficie algebriche reali, Annali di Mat. (3) 23, 215-283 (1915).
[12] M. Coste and M.-F. Roy, Thom’s lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets, J. Symbolic Comput. 5 (1988), no. 1-2, 121 – 129. · Zbl 0689.14006
[13] Bernard Deconinck and Mark van Hoeij, Computing Riemann matrices of algebraic curves, Phys. D 152/153 (2001), 28 – 46. Advances in nonlinear mathematics and science. · Zbl 1054.14079
[14] Bernard Deconinck, Matthias Heil, Alexander Bobenko, Mark van Hoeij, and Marcus Schmies, Computing Riemann theta functions, Math. Comp. 73 (2004), no. 247, 1417 – 1442. · Zbl 1092.33018
[15] Bernard Deconinck and Matthew S. Patterson, Computing with plane algebraic curves and Riemann surfaces: the algorithms of the Maple package ”algcurves”, Computational approach to Riemann surfaces, Lecture Notes in Math., vol. 2013, Springer, Heidelberg, 2011, pp. 67 – 123. · Zbl 1213.14114
[16] B.A. Dubrovin, Matrix finite-zone operators, Revs. Sci. Tech. 23, 20-50 (1983). · Zbl 0561.58043
[17] B. A. Dubrovin and S. M. Natanzon, Real theta-function solutions of the Kadomtsev-Petviashvili equation, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 2, 267 – 286, 446 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 2, 269 – 288. · Zbl 0672.35072
[18] H. Feng, Decomposition and Computation of the Topology of Plane Real Algebraic Curves, Ph.D. thesis, The Royal Institute of Technology, Stockholm (1992).
[19] J. Frauendiener and C. Klein, Hyperelliptic theta-functions and spectral methods, J. Comput. Appl. Math. 167 (2004), no. 1, 193 – 218. · Zbl 1052.65107
[20] J. Frauendiener and C. Klein, Hyperelliptic theta-functions and spectral methods: KdV and KP solutions, Lett. Math. Phys. 76 (2006), no. 2-3, 249 – 267. · Zbl 1127.14032
[21] Jörg Frauendiener and Christian Klein, Algebraic curves and Riemann surfaces in Matlab, Computational approach to Riemann surfaces, Lecture Notes in Math., vol. 2013, Springer, Heidelberg, 2011, pp. 125 – 162. · Zbl 1210.14070
[22] A. Gabard, Sur la topologie et la géométrie des courbes algébriques réelles, Ph.D. Thesis (2004). · Zbl 1141.14302
[23] Laureano Gonzalez-Vega and Ioana Necula, Efficient topology determination of implicitly defined algebraic plane curves, Comput. Aided Geom. Design 19 (2002), no. 9, 719 – 743. · Zbl 1043.68105
[24] D. A. Gudkov, Complete topological classification of the disposition of ovals of a sixth order curve in the projective plane, Gor\(^{\prime}\)kov. Gos. Univ. Učen. Zap. Vyp. 87 (1969), 118 – 153 (Russian).
[25] Axel Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), no. 2, 189 – 198 (German). · JFM 08.0317.04
[26] James L. Hafner and Kevin S. McCurley, Asymptotically fast triangularization of matrices over rings, SIAM J. Comput. 20 (1991), no. 6, 1068 – 1083. · Zbl 0738.68050
[27] David Hilbert, Ueber die reellen Züge algebraischer Curven, Math. Ann. 38 (1891), no. 1, 115 – 138 (German). · JFM 23.0753.02
[28] D. Hilbert, Mathematische Probleme, Arch. Math. Phys., 1:43-63, (German) (1901). · JFM 32.0084.05
[29] A. H. M. Hoare and D. Singerman, The orientability of subgroups of plane groups, Groups — St. Andrews 1981 (St. Andrews, 1981) London Math. Soc. Lecture Note Ser., vol. 71, Cambridge Univ. Press, Cambridge, 1982, pp. 221 – 227. · Zbl 0489.20036
[30] Hoon Hong, An efficient method for analyzing the topology of plane real algebraic curves, Math. Comput. Simulation 42 (1996), no. 4-6, 571 – 582. Symbolic computation, new trends and developments (Lille, 1993). · Zbl 1037.14503
[31] Caroline Kalla, New degeneration of Fay’s identity and its application to integrable systems, Int. Math. Res. Not. IMRN 18 (2013), 4170 – 4222. · Zbl 1316.35262
[32] C. Kalla and C. Klein, On the numerical evaluation of algebro-geometric solutions to integrable equations, Nonlinearity 25 (2012), no. 3, 569 – 596. · Zbl 1251.37067
[33] Felix Klein, On Riemann’s theory of algebraic functions and their integrals. A supplement to the usual treatises, Translated from the German by Frances Hardcastle, Dover Publications, Inc., New York, 1963.
[34] Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. · Zbl 0984.00001
[35] T. M. Malanyuk, Finite-gap solutions of the Davey-Stewartson equations, J. Nonlinear Sci. 4 (1994), no. 1, 1 – 21. · Zbl 0795.35117
[36] S. M. Natanzon, Geometry and algebra of real forms of complex curves, Math. Z. 243 (2003), no. 2, 391 – 407. · Zbl 1071.14521
[37] I. Petrowsky, On the topology of real plane algebraic curves, Ann. of Math. (2) 39 (1938), no. 1, 189 – 209. · Zbl 0018.27004
[38] K. Rohn, Die Maximalzahl und Anordnung der Ovale bei der ebenen Kurve 6. Ordnung und bei der Fläche 4. Ordnung, Math. Ann. 73 (1913), no. 2, 177 – 229 (German). · JFM 44.0738.05
[39] Takis Sakkalis, The topological configuration of a real algebraic curve, Bull. Austral. Math. Soc. 43 (1991), no. 1, 37 – 50. · Zbl 0716.14034
[40] Raimund Seidel and Nicola Wolpert, On the exact computation of the topology of real algebraic curves, Computational geometry (SCG’05), ACM, New York, 2005, pp. 107 – 115. · Zbl 1387.68276
[41] M. Seppälä and R. Silhol, Moduli spaces for real algebraic curves and real abelian varieties, Math. Z. 201 (1989), no. 2, 151 – 165. · Zbl 0645.14012
[42] C. L. Tretkoff and M. D. Tretkoff, Combinatorial group theory, Riemann surfaces and differential equations, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 467 – 519. · Zbl 0557.30036
[43] M. Trott, Applying Groebner basis to three problems in geometry, Mathematica in Education and Research 6 (1): 15-28 (1997).
[44] Victor Vinnikov, Selfadjoint determinantal representations of real plane curves, Math. Ann. 296 (1993), no. 3, 453 – 479. · Zbl 0789.14029
[45] O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187 – 200. · Zbl 0576.14031
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