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Computing Parametric Geometric Resolutions

  • Éric Schost

Abstract

 Given a polynomial system of n equations in n unknowns that depends on some parameters, we define the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters.

The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d n , where d is a bound on the degrees of the input system. Then we present a probabilistic algorithm to compute a parametric resolution. Its complexity is polynomial in the size of the output and in the complexity of evaluation of the input system. The probability of success is controlled by a quantity polynomial in the Bézout number.

We present several applications of this process, notably to computa- tions in the Jacobian of hyperelliptic curves and to questions of real geometry.

Keywords

Rational Function Generic Solution Parametric Geometric Input System Polynomial System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Éric Schost
    • 1
  1. 1.Laboratoire GAGE, École polytechnique, 91128 Palaiseau Cedex, France (e-mail: schost@gage.polytechnique.fr)FR

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