Relations between roots and coefficients, interpolation and application to system solving

  • Authors:

  • Bernard Mourrain;Olivier Ruatta

  • Affiliations:

  • INRIA, GALAAD, BP 93, 06902 Sophia-Antipolis, France;INRIA, GALAAD, BP 93, 06902 Sophia-Antipolis, France

  • Venue:
  • Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
  • Year:
  • 2002

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Abstract

We propose an algebraic framework to represent zero-dimensional algebraic systems. In this framework, we give new interpolation formulæ. We use this good representation of the algebraic systems to develop a generalization of Weierstrass's method to the multivariate systems. This method allows us to approximate simultaneously all the roots of an algebraic system. We obtain an effective iteration function with a quadratic convergence in a neighbourhood of the solutions. We use this Weierstrass iteration function in a continuation method to obtain a global method. Experiments are exposed to underline the efficiency of the approach.