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A Symbolic-Numeric Algorithm for Genus Computation

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Numerical and Symbolic Scientific Computing

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

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Abstract

We report on a method for computing the genus of a plane complex algebraic curve based on the topology of singular points and on knot theory. We propose a symbolic-numeric algorithm to be used for plane complex algebraic curves whose defining polynomials have numeric coefficients. Together with its main functionality to compute the genus, the algorithm provides also tools for computational operations in knot theory. We split the main algorithm into several interdependent subalgorithms. We base some of our subalgorithms on general algorithms from computational geometry (e.g. Bentley-Ottman). Whenever required, we design our own subalgorithms for solving the specific problems (e.g. computation of the Alexander polynomial). We use for the implementation the Axel algebraic geometric modeler, developed at INRIA, Sophia-Antipolis.

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Acknowledgements

Many thanks to Bernard Mourrain who also contributed to the implementation of GENOM3CK and offered important computational and mathematical support and guidance whenever required. Many thanks to Julien Wintz, who contributed to the implementation of the library in its starting phase. We would like to especially thank Esther Klann and Ronny Ramlau, and the other colleagues from the “Doctoral Program-Computational Mathematics” for their helpful discussions and comments, which contributed with many useful insights to handling the numerical part of our problem.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Altenbergerstr. 69, 4040, Linz, Austria

    Mădălina Hodorog & Josef Schicho

Authors
  1. Mădălina Hodorog
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  2. Josef Schicho
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Correspondence to Mădălina Hodorog .

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Editors and Affiliations

  1. Inst. Numerische Mathematik, Universität Linz, Altenbergerstr. 69, Linz, 4040, Austria

    Ulrich Langer

  2. , Research Institute for Symbolic Computat, Johannes Kepler University, Altenbergerstr. 69, Linz, 4040, Austria

    Peter Paule

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Hodorog, M., Schicho, J. (2012). A Symbolic-Numeric Algorithm for Genus Computation. In: Langer, U., Paule, P. (eds) Numerical and Symbolic Scientific Computing. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0794-2_4

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  • DOI: https://doi.org/10.1007/978-3-7091-0794-2_4

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