Comprehensive Involutive Systems

  • Vladimir Gerdt
  • Amir Hashemi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)


In this paper we consider parametric ideals and introduce a notion of comprehensive involutive system. This notion plays the same role in theory of involutive bases as the notion of comprehensive Gröbner system in theory of Gröbner bases. Given a parametric ideal, the space of parameters is decomposed into a finite set of cells. Each cell yields the corresponding involutive basis of the ideal for the values of parameters in that cell. Using the Gerdt–Blinkov algorithm described in [6] for computing involutive bases and also the Montes DisPGB algorithm for computing comprehensive Gröbner systems [13], we present an algorithm for construction of comprehensive involutive systems. The proposed algorithm has been implemented in Maple, and we provide an illustrative example showing the step-by-step construction of comprehensive involutive system by our algorithm.


Symbolic Computation Polynomial Ideal Parametric Ideal Parametric Polynomial Involutive Basis 
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.
    Apel, J., Hemmecke, R.: Detecting unnecessary reductions in an involutive basis computation. J. Symbolic Computation 40, 1131–1149 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Becker, T., Weispfenning, T.: Gröbner Bases: a Computational Approach to Commutative Algebra. Graduate Texts in Mathematics, vol. 141. Springer, New York (1993)zbMATHGoogle Scholar
  3. 3.
    Buchberger, B.: Ein Algorithms zum Auffinden der Basiselemente des Restklassenrings nach einem nuildimensionalen Polynomideal. PhD thesis, Universität Innsbruck (1965)Google Scholar
  4. 4.
    Buchberger, B.: A Criterion for Detecting Unnecessary Reductions in the Cconstruction of Gröbner Bases. In: Ng, K.W. (ed.) EUROSAM 1979. LNCS, vol. 72, pp. 3–21. Springer, Heidelberg (1979)CrossRefGoogle Scholar
  5. 5.
    Buchberger, B., Winkler, F. (eds.): Gröbner Bases and Applications. London Mathematical Society Lecture Note Series, vol. 251. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  6. 6.
    Gerdt, V.P.: Involutive algorithms for computing Gröbner bases. In: Cojocaru, S., Pfister, G., Ufnarovski, V. (eds.) Computational Commutative and Non-Commutative Algebraic Geometry, pp. 199–225. IOS Press, Amstrerdam (2005) (arXiv:math/0501111)Google Scholar
  7. 7.
    Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Mathematics and Computers in Simulation 45, 519–542 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gerdt, V.P., Blinkov, Y.A.: Involutive Division Generated by an Antigraded Monomial Ordering. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 158–174. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Janet, M.: Les Systèmes d’Équations aux Dérivées Partielles. Journal de Mathématique 3, 65–151 (1920)Google Scholar
  10. 10.
    Kapur, D., Sun, Y., Wand, D.: A new algorithm for computing comprehensive Gröbner systems. In: Watt, S.M. (ed.) Proc. ISSAC 2010, pp. 29–36. ACM Press, New York (2010)Google Scholar
  11. 11.
    Manubens, M., Montes, A.: Improving DisPGB algorithm using the discriminant ideal. J. Symbolic Computation 41, 1245–1263 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Manubens, M., Montes, A.: Minimal canonical comprehensive Gröbner systems. J. Symbolic Computation 44, 463–478 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Montes, A.: A new algorithm for discussing Gröbner bases with parameters. J. Symbolic Computation 33, 183–208 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Montes, A.: Solving the load flow problem using Gröbner bases. SIGSAM Bulletin 29, 1–13 (1995)CrossRefGoogle Scholar
  15. 15.
    Montes, A., Wibmer, M.: Gröbner bases for polynomial systems with parameters. J. Symbolic Computation 45, 1391–1425 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Pommaret, J.-F.: Systems of Partial Differential Equations and Lie Pseudogroups. Mathematics and its Applications, vol. 14. Gordon & Breach Science Publishers, New York (1978)zbMATHGoogle Scholar
  17. 17.
    Sato, Y., Suzuki, A.: An alternative approach to comprehensive Gröbner bases. J. Symbolic Computation 36, 649–667 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Sato, Y., Suzuki, A.: A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In: Trager, B.M. (ed.) Proc. ISSAC 2006, pp. 326–331. ACM Press, New York (2006)Google Scholar
  19. 19.
    Suzuki, A.: Computation of Full Comprehensive Gröbner Bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 431–444. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Seiler, W.M.: Involution - The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics, vol. 24. Springer, Berlin (2010)zbMATHGoogle Scholar
  21. 21.
    Weispfenning, V.: Cannonical comprehensive Gröbner bases. J. Symbolic Computation 36, 669–683 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Weispfenning, V.: Comprehensive Gröbner bases. J. Symbolic Computation 14, 1–29 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Zharkov, A.Y., Blinkov, Y.A.: Involutive approach to investigating polynomial systems. Mathematics and Computers in Simulation 42, 323–332 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Vladimir Gerdt
    • 1
  • Amir Hashemi
    • 2
  1. 1.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubnaRussia
  2. 2.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

Personalised recommendations