On the geometry of polar varieties

  • Bernd Bank
  • Marc Giusti
  • Joos Heintz
  • Mohab Safey El Din
  • Eric Schost


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.


Real polynomial equation solving Singularities Classic polar varieties Dual polar varieties Generic polar varieties Meagerly generic polar varieties 

Mathematics Subject Classification (2000)

14P05 14B05 14Q10 14Q15 68W30 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Bernd Bank
    • 1
  • Marc Giusti
    • 2
  • Joos Heintz
    • 3
    • 4
  • Mohab Safey El Din
    • 5
  • Eric Schost
    • 6
  1. 1.Humboldt-Universität zu Berlin, Institut für MathematikBerlinGermany
  2. 2.CNRS, École Polytechnique, Laboratoire LIXPalaiseau CedexFrance
  3. 3.Departamento de ComputaciónUniversidad de Buenos Aires and CONICET, Ciudad Univ., Pab.IBuenos AiresArgentina
  4. 4.Departamento de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain
  5. 5.UPMC, Univ Paris 06, INRIA, Paris-Rocquencourt, SALSA Project; LIP6; CNRS, UMR 7606, LIP6; UFR Ingéniérie 919, LIP6 Passy–KennedyParisFrance
  6. 6.Computer Science Department, Room 415, Middlesex CollegeUniversity of Western OntarioLondonCanada

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