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Comprehensive Triangular Decomposition

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Computer Algebra in Scientific Computing (CASC 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4770))

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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.

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Authors and Affiliations

  1. University of Western Ontario, London N6A 1M8, Canada

    Changbo Chen, Oleg Golubitsky, Marc Moreno Maza & Wei Pan

  2. Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France

    François Lemaire

Authors
  1. Changbo Chen
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  2. Oleg Golubitsky
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  3. François Lemaire
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  4. Marc Moreno Maza
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  5. Wei Pan
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Editor information

Victor G. Ganzha Ernst W. Mayr Evgenii V. Vorozhtsov

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© 2007 Springer-Verlag Berlin Heidelberg

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Chen, C., Golubitsky, O., Lemaire, F., Maza, M.M., Pan, W. (2007). Comprehensive Triangular Decomposition. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2007. Lecture Notes in Computer Science, vol 4770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75187-8_7

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  • DOI: https://doi.org/10.1007/978-3-540-75187-8_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-75186-1

  • Online ISBN: 978-3-540-75187-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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