On the geometry of polar varieties

  • Authors:

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

  • Affiliations:

  • Humboldt-Universitä/t zu Berlin, Institut fü/r Mathematik, 10099, Berlin, Germany;CNRS, É/cole Polytechnique, Laboratoire LIX, 91228, Palaiseau Cedex, France;Univ. de Buenos Aires and CONICET, Ciudad Univ., Pab.I, Depo. de Computació/n, Buenos Aires, Argentina and Univ. de Cantabria, Depo. de Matemá/ticas, Estadí/stica y Computaci� ...;UPMC, Univ Paris 06, INRIA, Paris-Rocquencourt, SALSA Project/ LIP6/ CNRS, UMR 7606, LIP6/ UFR Ingé/nié/rie 919, LIP6 Passy–/Kennedy, Case 169, 4, Place Jussieu, 75252, Paris, Fr ...;University of Western Ontario, Computer Science Department, Room 415, Middlesex College, London, ON, Canada

  • Venue:
  • Applicable Algebra in Engineering, Communication and Computing
  • Year:
  • 2010

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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.