Generalized polar varieties: geometry and algorithms

  • Authors:

  • B. Bank;M. Giusti;J. Heintz;L. M. Pardo

  • Affiliations:

  • Humboldt-Universität zu Berlin, Institut für Mathematik, Berlin, Germany;Laboratoire STIX, École Polytechnique, Palaiseau Cedex, France;Depo. de Computación, Fac. de Ciencias Exactas y Naturales, Univ. de Buenos Aires and CONICET, Argentina and Depo. de Matemáticas, Estadística y Computación, Fac. de Ciencias, ...;Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Santander, Spain

  • Venue:
  • Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
  • Year:
  • 2005

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Abstract

Let V be a closed algebraic subvariety of the n-dimensional projective space over the complex or real numbers and suppose that V is non-empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in Bank et al. (Kybernetika 40 (2004), to appear). As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar variety, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in the context of algorithmic elimination theory a highly efficient, probabilistic elimination procedure for the following task: In case, that the variety V is Q-definable and affine, having a complete intersection ideal of definition, and that the real trace of V is non-empty and smooth, find for each connected component of the real trace of V an algebraic sample point.