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On the Topology of Real Algebraic Plane Curves

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Abstract

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.

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Authors and Affiliations

  1. KLMM, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing, China

    Jinsan Cheng

  2. INRIA Nancy Grand Est, LORIA Laboratory, Nancy, France

    Sylvain Lazard, Luis Peñaranda & Marc Pouget

  3. INRIA Paris-Rocquencourt, Paris, France

    Fabrice Rouillier

  4. LIP6 (Université Paris 6, CNRS), Paris, France

    Fabrice Rouillier

  5. Department of Computer Science, Aarhus University, Aarhus, Denmark

    Elias Tsigaridas

Authors
  1. Jinsan Cheng
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  2. Sylvain Lazard
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  3. Luis Peñaranda
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  4. Marc Pouget
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  5. Fabrice Rouillier
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  6. Elias Tsigaridas
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Correspondence to Marc Pouget.

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Cheng, J., Lazard, S., Peñaranda, L. et al. On the Topology of Real Algebraic Plane Curves. Math.Comput.Sci. 4, 113–137 (2010). https://doi.org/10.1007/s11786-010-0044-3

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  • DOI: https://doi.org/10.1007/s11786-010-0044-3

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