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Algebraic Curves and Riemann Surfaces in Matlab

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Computational Approach to Riemann Surfaces

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2013))

Abstract

In the previous chapter, a detailed description of the algorithms for the ‘algcurves’ package in Maple was presented. As discussed there, the package is able to handle general algebraic curves with coefficients given as exact arithmetic expressions, a restriction due to the use of exact integer arithmetic. Coefficients in terms of floating point numbers, i.e., the representation of decimal numbers of finite length on a computer, can in principle be handled, but the floating point numbers have to be converted to rational numbers. This can lead to technical difficulties in practice. One also faces limitations if one wants to study families of Riemann surfaces, where the coefficients in the algebraic equation defining the curve are floating point numbers depending on a set of parameters, i.e., if one wants to explore modular properties of Riemann surfaces as in the examples discussed below. An additional problem in this context can be computing time since the computation of the Riemann matrix uses the somewhat slow Maple integration routing. Thus, a more efficient computation of the Riemann matrix is interesting if one wants to study families of Riemann surfaces or higher genus examples which are computationally expensive.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, 56, Dunedin, 9010, New Zealand

    Jörg Frauendiener

  2. Centre of Mathematics for Applications, University of Oslo, 1053, Blindern, NO-0316, Oslo, Norway

    Jörg Frauendiener

  3. Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078, Dijon Cedex, France

    Christian Klein

Authors
  1. Jörg Frauendiener
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  2. Christian Klein
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Correspondence to Jörg Frauendiener .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Technical University of Berlin, Straße des 17.Juni 136, Berlin, 10623, Germany

    Alexander I. Bobenko

  2. Institut de Mathématiques, Université de Bourgogne, av. Alain Savary 9, Dijon Cedex, 21078, France

    Christian Klein

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Frauendiener, J., Klein, C. (2011). Algebraic Curves and Riemann Surfaces in Matlab. In: Bobenko, A., Klein, C. (eds) Computational Approach to Riemann Surfaces. Lecture Notes in Mathematics(), vol 2013. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17413-1_3

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