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Comprehensive Triangular Decomposition

  • Changbo Chen
  • Oleg Golubitsky
  • François Lemaire
  • Marc Moreno Maza
  • Wei Pan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4770)

Abstract

We introduce the concept of comprehensive triangular decomposition (CTD) for a parametric polynomial system F with coefficients in a field. In broad words, this is a finite partition of the the parameter space into regions, so that within each region the “geometry” (number of irreducible components together with their dimensions and degrees) of the algebraic variety of the specialized system F(u) is the same for all values u of the parameters.

We propose an algorithm for computing the CTD of F. It relies on a procedure for solving the following set theoretical instance of the coprime factorization problem. Given a family of constructible sets A 1, ..., A s , compute a family B 1, ..., B t of pairwise disjoint constructible sets, such that for all 1 ≤ i ≤ s the set A i writes as a union of some of the B 1, ..., B t .

We report on an implementation of our algorithm computing CTDs, based on the RegularChains library in maple. We provide comparative benchmarks with maple implementations of related methods for solving parametric polynomial systems. Our results illustrate the good performances of our CTD code.

Keywords

Algebraic Variety Polynomial System Recursive Call Regular System Triangular Decomposition 
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 2007

Authors and Affiliations

  • Changbo Chen
    • 1
  • Oleg Golubitsky
    • 1
  • François Lemaire
    • 2
  • Marc Moreno Maza
    • 1
  • Wei Pan
    • 1
  1. 1.University of Western Ontario, London N6A 1M8Canada
  2. 2.Université de Lille 1, 59655 Villeneuve d’Ascq CedexFrance

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