On Connection Between Constructive Involutive Divisions and Monomial Orderings

  • Alexander Semenov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4194)


This work considers the basic issues of the theory of involutive divisions, namely, the property of constructivity which assures the existence of minimal involutive basis. The work deals with class of ≻ -divisions which possess many good properties of Janet division and can be considered as its analogs for orderings different from the lexicographic one. Various criteria of constructivity and non-constructivity are given in the paper for these divisions in terms of admissible monomial orderings ≻ . It is proven that Janet division has the advantage in the minimal involutive basis size of the class of ≻ -divisions for which x 1x 2 ≻ ... ≻ x n holds. Also examples of new involutive divisions which can be better than Janet division in minimal involutive basis size for some ideals are given.


Distinct Element Polynomial Ideal Monomial Ideal Monomial Ordering Admissible Ordering 
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.: The theory of involutive divisions and an application to Hilbert function computations. Journal of Symbolic Computation 25(6), 683–704 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Calmet, J., Hausdorf, M., Seiler, W.M.: A constructive introduction to involution. In: Proc. Int. Symp. Applications of Computer Algebra – ISACA 2000, pp. 33–50 (2001)Google Scholar
  3. 3.
    Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Mathematics and Computers in Simulation 45, 519–542 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gerdt, V.P., Blinkov, Y.A.: Minimal involutive bases. Mathematics and Computers in Simulation 45, 543–560 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gerdt, V.P.: Involutive division technique: some generalizations and optimizations. Journal of Mathematical Sciences 108(6), 1034–1051 (2002)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Hemmecke, R.: Involutive Bases for Polynomial Ideals. Institut für Symbolisches Rechnen, Linz (2003)Google Scholar
  7. 7.
    Semenov, A.S.: Pair analysis of involutive divisions. Fundamental and Applied Mathematics 9(3), 199–212 (2003)zbMATHGoogle Scholar
  8. 8.
    Gerdt, V.P., Blinkov, Y.A., Yanovich, D.A.: Quick search of Janet divisor. Programmirovanie 1, 32–36 (2001) (in Russian)MathSciNetGoogle Scholar
  9. 9.
    Zharkov, A.Y., Blinkov, Y.A.: Involutive systems of algebraic equations. Programmirovanie (1994) (in Russian)Google Scholar
  10. 10.
    Zharkov, A.Y., Blinkov, Y.A.: Involutive Bases of Zero-Dimensional Ideals. Preprint No. E5-94-318, Joint Institute for Nuclear Research, Dubna (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexander Semenov
    • 1
  1. 1.Department of Mechanics and MathematicsMoscow State University 

Personalised recommendations