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Polars of Real Singular Plane Curves

  • Heidi Camilla Mork
  • Ragni Piene
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)

Abstract

Polar varieties have in recent years been used by Bank, Giusti, Heintz, Mbakop, and Pardo, and by Safey El Din and Schost, to find efficient procedures for determining points on all real components of a given non-singular algebraic variety. In this note we review the classical notion of polars and polar varieties, as well as the construction of what we here call reciprocal polar varieties. In particular we consider the case of real affine plane curves, and we give conditions for when the polar varieties of singular curves contain points on all real components.

Key words

Polar varieties hypersurfaces plane curves tangent space flag 

AMS(MOS) subject classifications

Primary 14H50 Secondary 14J70 14P05 14Q05 

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References

  1. au][1]
    B. BANK, M. GIUSTI, J. HEINTZ, AND G.M. MBAKOP. Polar varieties, real equation solving, and data structures: The hypersurface case. J. Complexity, 13(1): 5–27, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    B. BANK, M. GIUSTI, J. HEINTZ, AND G.M. MBAKOP. Polar varieties and efficient real elimination. Math. Z., 238(1): 115–144, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    B. BANK, M. GIUSTI, J. HEINTZ, AND L.M. PARDO. Generalized polar varieties and an efficient real elimination procedure. Kybernetika (Prague), 40(5): 519–550, 2004.MathSciNetGoogle Scholar
  4. [4]
    B. BANK, M. GIUSTI, J. HEINTZ, AND L.M. PARDO. Generalized polar varieties: Geometry and algorithms. J. Complexity, 21(4): 377–412, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    S. ENDRASS et al. Surf 1.0.4. 2003. A Computer Software for Visualising Real Algebraic Geometry, http://surf.sourceforge.net.Google Scholar
  6. [6]
    R. PIENE. Polar classes of singular varieties. Ann. Sci. École Norm. Sup. (4), 11(2): 247–276, 1978.zbMATHMathSciNetGoogle Scholar
  7. [7]
    M. SAFEY EL DIN. Putting into practice deformation techniques for the computation of sampling points in real singular hypersurfaces. 2005.Google Scholar
  8. [8]
    M. SAFEY EL DIN AND E. SCHOST. Polar varieties and computation of one point in each connected component of a smooth algebraic set. In Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp. 224–231 (electronic), New York, 2003. ACM.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Heidi Camilla Mork
    • 1
  • Ragni Piene
    • 1
  1. 1.CMA, Department of MathematicsUniversity of OsloOsloNorway

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