Involutive Division Generated by an Antigraded Monomial Ordering

  • Vladimir P. Gerdt
  • Yuri A. Blinkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6885)


In the present paper we consider a class of involutive monomial divisions pairwise constructed by the partition of variables into multiplicative and nonmultiplicative generated by a total monomial ordering. If this ordering is admissible or the inverse of an admissible ordering, then the involutive division generated possesses all algorithmically important properties such as continuity, constructivity, and noetherianity. Among all such divisions, we single out those generated by antigraded monomial orderings. We demonstrate, by example of the antigraded lexicographic ordering, that the divisions of this class are heuristically better than the classical Janet division. The last division is pairwise generated by the pure lexicographic ordering and up to now has been considered as computationally best.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Mathematics and Computers in Simulation 45, 519–542 (1998); Minimal involutive bases, ibid, 543–560MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Apel, J.: The theory of involutive divisions and an application to Hilbert function computations. J. Symbolic Computation 25, 683–704 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Janet, M.: Leçons sur les Systèmes d’Equations aux Dérivées Partielles. Cahiers Scientifiques, IV, Gauthier-Villars, Paris (1929)Google Scholar
  4. 4.
    Gerdt, V.P.: Involutive algorithms for computing Gröbner bases. In: Computational Commutative and Non-Commutative Algebraic Geometry, pp. 199–225. IOS Press, Amsterdam (2005)Google Scholar
  5. 5.
    Seiler, W.M.: Involution: The formal theory of differential equations and its applications in computer algebra. In: Algorithms and Computation in Mathematics, vol. 24. Springer, Heidelberg (2010)Google Scholar
  6. 6.
    Gerdt, V.P., Blinkov, Y. A.: Specialized computer algebra system GINV. Programming and Computer Software 34(2), 112–123 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
  8. 8.
    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-2 - A computer algebra system for polynomial computations (2010),
  9. 9.
  10. 10.
  11. 11.
    Gerdt, V.P.: Involutive division technique: some generalizations and optimizations. J. Math. Sciences 108(6), 1034–1051 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, Y.-F., Gao, X.-S.: Involutive directions and new involutive divisions. Computers and Mathematics with Applications 41, 945–956 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Semenov, A.S.: On connection between constructive involutive divisions and monomial orderings. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 261–278. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Semenov, A.S.: Constructivity of involutive divisions. Programming and Computer Software 32(2), 96–102 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Semenov, A.S., Zyuzikov, P.A.: Involutive divisions and monomial orderings. Programming and Computer Software 33(3), 139–146 (2007); Involutive divisions and monomial orderings: Part II. Ibid 34(2), 107–111 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry, 2nd edn. Graduate Texts in Mathematics, vol. 185. Springer, New York (2005)zbMATHGoogle Scholar
  17. 17.
    Greul, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer, Berlin (2007)Google Scholar
  18. 18.
    Becker, T., Weispfenning, V.: Gröbner Bases. A Computational Approach to Commutative Algebra. Graduate Texts in Mathematics, vol. 141. Springer, New York (1993)zbMATHGoogle Scholar
  19. 19.
    Gerdt, V.P.: On the relation between Pommaret and Janet bases. In: Computer Algebra in Scientific Computing / CASC 2000, pp. 167–181. Springer, Berlin (2000)Google Scholar
  20. 20.
    Bächler, T., Gerdt, V.P., Lange-Hegermann, M., Robertz, D.: Thomas decomposition of algebraic and differential systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2010. LNCS, vol. 6244, pp. 31–54. Springer, Heidelberg (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Vladimir P. Gerdt
    • 1
  • Yuri A. Blinkov
    • 2
  1. 1.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubnaRussia
  2. 2.Department of Mathematics and MechanicsSaratov State UniversitySaratovRussia

Personalised recommendations