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On the geometry of polar varieties

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Abstract

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non–emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment.

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Authors and Affiliations

  1. Humboldt-Universität zu Berlin, Institut für Mathematik, 10099, Berlin, Germany

    Bernd Bank

  2. CNRS, École Polytechnique, Laboratoire LIX, 91228, Palaiseau Cedex, France

    Marc Giusti

  3. Departamento de Computación, Universidad de Buenos Aires and CONICET, Ciudad Univ., Pab.I, 1428 Ciudad Autónoma de Buenos Aires, Buenos Aires, Argentina

    Joos Heintz

  4. Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39071, Santander, Spain

    Joos Heintz

  5. UPMC, Univ Paris 06, INRIA, Paris-Rocquencourt, SALSA Project; LIP6; CNRS, UMR 7606, LIP6; UFR Ingéniérie 919, LIP6 Passy–Kennedy, Case 169, 4, Place Jussieu, 75252, Paris, France

    Mohab Safey El Din

  6. Computer Science Department, Room 415, Middlesex College, University of Western Ontario, London, ON, Canada

    Eric Schost

Authors
  1. Bernd Bank
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  2. Marc Giusti
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  3. Joos Heintz
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  4. Mohab Safey El Din
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  5. Eric Schost
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Corresponding author

Correspondence to Bernd Bank.

Additional information

Research partially supported by the following Argentinian, Canadian, French and Spanish agencies and grants: UBACYT X-098, UBACYT X-113, PICT–2006–02067, the Canada Research Chair Program, NSERC, BLAN NT05-4-45732 (projet GECKO), MTM 2007-62799.

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Bank, B., Giusti, M., Heintz, J. et al. On the geometry of polar varieties. AAECC 21, 33–83 (2010). https://doi.org/10.1007/s00200-009-0117-1

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  • DOI: https://doi.org/10.1007/s00200-009-0117-1

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