A New Definition for Passivity and Its Relation to Coherence

  • Moritz Minzlaff
  • Jacques Calmet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4120)


It is an essential step in decomposition algorithms for radical differential ideals to satisfy the so-called Rosenfeld property. Basically all approaches to achieve this step are based on one of two concepts: Coherence or passivity.

In this paper we will give a modern treatment of passivity. Our focus is on questions regarding the different definitions of passivity and their relation to coherence. The theorem by Li and Wang stating that passivity in Wu’s sense implies coherence is extended to a broader setting. A new definition for passivity is suggested and it is shown to allow for a converse statement so that coherence and passivity are seen to be equivalent.


Induction Hypothesis Algebraic Group Decomposition Algorithm Derivative Operator Polynomial Algebra 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Computing representations for radicals of finitely generated differential ideals. Technical report, no. IT306, LIFL (1997)Google Scholar
  2. 2.
    Chen, Y.: Private communication (2005)Google Scholar
  3. 3.
    Chen, Y., Gao, X.: Involutive Characteristic Sets of Algebraic Partial Differential Equation Systems. Science in China (Series A) 46(4), 469–487 (2003)MathSciNetGoogle Scholar
  4. 4.
    Gerdt, V.P., Blinkov, Y.A.: Involutive Bases of Polynomial Ideals. Mathematics and Computers in Simulation 45, 519 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Hubert, E.: Factorization Free Decomposition Algorithms in Differential Algebra. Journal of Symbolic Computation 29(4–5), 622–641 (2000)MathSciNetGoogle Scholar
  6. 6.
    Hubert, E.: Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems. In: Winkler, F., Langer, U. (eds.) SNSC 2001. LNCS, vol. 2630, pp. 40–87. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Hubert, E., Le Roux, N.: Computing power series solutions of a nonlinear PDE system. In: ISSAC 2003: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp. 148–155 (2003)Google Scholar
  8. 8.
    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Pure and Applied Mathematics 54 (1973)Google Scholar
  9. 9.
    Li, Z., Wang, D.: Coherent, Regular and Simple Systems in Zero Decompositions of Partial Differential Systems. Systems Science and Mathematical Sciences 12(5), 43–60 (1999)zbMATHGoogle Scholar
  10. 10.
    Minzlaff, M.: On The Decomposition of Radical Differential Ideals, Diplomarbeit, Universität Karlsruhe (TH) (2006)Google Scholar
  11. 11.
    Ritt, J.F.: Differential Algebra. Amer. Math. Soc. 33 (1950)Google Scholar
  12. 12.
    Rosenfeld, A.: Specializations in Differential Algebra. Trans. Amer. Math. Soc. 90, 394–407 (1959)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Seiler, W.M.: A Combinatorial Approach to Involution and Delta-Regularity I: Involutive Bases in Polynomial Algebras of Solvable Type. Preprint Universität Mannheim (2002)Google Scholar
  14. 14.
    Sit, W.: The Ritt-Kolchin theory for differential polynomials. Differential Algebra and Related Topics, 1–70 (2002)Google Scholar
  15. 15.
    Wu, W.T.: On The Foundation of Algebraic Differential Geometry. Systems Science and Mathematical Sciences 2(4), 289–312 (1989)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Moritz Minzlaff
    • 1
  • Jacques Calmet
    • 2
  1. 1.Universität Karlsruhe 
  2. 2.Fakultät für InformatikUniversität Karlsruhe 

Personalised recommendations