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A Symbolic-Numeric Algorithm for Genus Computation

  • Mădălina HodorogEmail author
  • Josef Schicho
Chapter
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Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

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.

Keywords

Symbolic-numeric Algorithms Complex Algebraic Plane Curves Alexander Polynomial Algebraic Link General Computer Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

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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Copyright information

© Springer-Verlag/Wien 2012

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied MathematicsLinzAustria

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