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Mathematische Zeitschrift

, Volume 238, Issue 1, pp 115–144 | Cite as

Polar varieties and efficient real elimination

  • B. Bank
  • M. Giusti
  • J. Heintz
  • G.M. Mbakop
Original article

Abstract.

Let \(S_0\) be a smooth and compact real variety given by a reduced regular sequence of polynomials \(f_1, \ldots, f_p\). This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of \(S_0\) . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of \(S_0\). This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations \(f_1,\ldots, f_p\) and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system \(f_1,\ldots,f_p\).

Mathematics Subject Classification (1991): 14P05, 14B05, 68W30 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • B. Bank
    • 1
  • M. Giusti
    • 2
  • J. Heintz
    • 3
  • G.M. Mbakop
    • 1
  1. 1.Institut für Mathematik, Humboldt-Universität zu Berlin, Germany (e-mail: bank@mathematik.hu-berlin.de, mbakop@mathematik.hu-berlin.de) DE
  2. 2.UMS MEDICIS, Laboratoire GAGE, École Polytechnique, Palaiseau, France (e-mail: giusti@gage.polytechnique.fr) FR
  3. 3.Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Santander, Spain (e-mail: heintz@matesco.unican.es) ES

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