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)


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.


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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  1. 1.
    Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comp. 28(1-2), 105–124 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bernstein, D.J.: Factoring into coprimes in essentially linear time. J. Algorithms 54(1), 1–30 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boulier, F., Lemaire, F., Moreno Maza, M.: Well known theorems on triangular systems and the D5 principle. In: Proc. of Transgressive Computing 2006, Granada, Spain (2006)Google Scholar
  4. 4.
    Caviness, B., Johnson, J. (eds.): Quantifier Elimination and Cylindical Algebraic Decomposition, Texts and Mongraphs in Symbolic Computation. Springer, Heidelberg (1998)Google Scholar
  5. 5.
    Chen, F., Wang, D. (eds.): Geometric Computation. Lecture Notes Series on Computing, vol. 11. World Scientific Publishing Co, Singapore, New Jersey (2004)zbMATHGoogle Scholar
  6. 6.
    Chou, S.C., Gao, X.S.: Computations with parametric equations. In: Proc. ISAAC 1991, Bonn, Germany, pp. 122–127 (1991)Google Scholar
  7. 7.
    Chou, S.C., Gao, X.S.: Solving parametric algebraic systems. In: Proc. ISSAC 1992, Berkeley, California, pp. 335–341 (1992)Google Scholar
  8. 8.
    Dahan, X., Moreno Maza, M., Schost, É., Xie, Y.: On the complexity of the D5 principle. In: Proc. of Transgressive Computing 2006, Granada, Spain (2006)Google Scholar
  9. 9.
    Duval, D.: Algebraic Numbers: an Example of Dynamic Evaluation. J. Symb. Comp. 18(5), 429–446 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gómez Díaz, T.: Quelques applications de l’évaluation dynamique. PhD thesis, Université de Limoges (1994)Google Scholar
  11. 11.
    Kalkbrener, M.: A generalized euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comp. 15, 143–167 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lazard, D., Rouillier, F.: Solving parametric polynomial systems. Technical Report 5322, INRIA (2004)Google Scholar
  13. 13.
    Manubens, M., Montes, A.: Improving dispgb algorithm using the discriminant ideal (2006)Google Scholar
  14. 14.
    Montes, A.: A new algorithm for discussing gröbner bases with parameters. J. Symb. Comput. 33(2), 183–208 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Moreno Maza, M.: On triangular decompositions of algebraic varieties. Technical Report TR 4/99, NAG Ltd, Oxford, UK (1999),
  16. 16.
    Samuel, P., Zariski, O.: Commutative algebra. D. Van Nostrand Company, INC (1967)Google Scholar
  17. 17.
    Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gröbner bases. In: Proc. ISSAC 2006, pp. 326–331. ACM Press, New York (2006)CrossRefGoogle Scholar
  18. 18.
    The SymbolicData Project (2000–2006),
  19. 19.
    Wang, D.M.: Computing triangular systems and regular systems. Journal of Symbolic Computation 30(2), 221–236 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Wang, D.M.: Decomposing polynomial systems into simple systems. J. Symb. Comp. 25(3), 295–314 (1998)zbMATHCrossRefGoogle Scholar
  21. 21.
    Wang, D.M.: Elimination Methods. Springer, Wein, New York (2000)Google Scholar
  22. 22.
    Weispfenning, V.: Comprehensive grobner bases. J. Symb. Comp. 14, 1–29 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Weispfenning, V.: Canonical comprehensive grobner bases. In: ISSAC 2002, pp. 270–276. ACM Press, New York (2002)CrossRefGoogle Scholar
  24. 24.
    Wu, W.T.: A zero structure theorem for polynomial equations solving. MM Research Preprints 1, 2–12 (1987)Google Scholar
  25. 25.
    Wu, W.T.: On a projection theorem of quasi-varieties in elimination theory. MM Research Preprints 4, 40–53 (1989)Google Scholar
  26. 26.
    Yang, L., Hou, X.R., Xia, B.C.: A complete algorithm for automated discovering of a class of inequality-type theorem. Science in China, Series E 44(6), 33–49 (2001)zbMATHMathSciNetGoogle Scholar

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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