# Dynamical Aspects of Involutive Bases Computations

• Ralf Hemmecke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)

## Abstract

The article is a contribution to a more efficient computation of involutive bases. We present an algorithm which computes a ‘sliced division’. A sliced division is an admissible partial division in the sense of Apel. Admissibility requires a certain order on the terms. Instead of ordering the terms in advance, our algorithm additionally returns such an order for which the computed sliced division is admissible. Our algorithm gives rise to a whole class of sliced divisions since there is some freedom to choose certain elements in the course of its run. We show that each sliced division refines the Thomas division and thus leads to terminating completion algorithms for the computation of involutive bases. A sliced division is such that its cones ‘cover’ a relatively ‘big’ part of the term monoid generated by the given terms. The number of prolongations that must be considered during the involutive basis algorithm is tightly connected to the dimensions and number of the cones. By some computer experiments, we show how this new division can be fruitful for the involutive basis algorithm.

We generalise the sliced division algorithm so that it can be seen as an algorithm which is parameterised by two choice functions and give particular choice functions for the computation of the classical divisions of Janet, Pommaret, and Thomas.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Joachim Apel. The theory of involutive divisions and an application to Hilbert function computations. Journal of Symbolic Computation, 25(6):683–704, June 1998.
2. 2.
Joachim Apel. Zu Berechenbarkeitsfragen der Idealtheorie. Habilitationsschrift, Universität Leipzig, Fakultät für Mathematik und Informatik, Augustusplatz 10-11, 04109 Leipzig, 1998.Google Scholar
3. 3.
Bruno Buchberger. Gröbner bases: An algorithmic method in polynomial ideal theory. In N. K. Bose, editor, Recent Trends in Multidimensional Systems Theory, chapter 6, pages 184–232. D. Reidel Publishing Company, Dordrecht, The Netherlands, 1985.Google Scholar
4. 4.
Vladimir P. Gerdt and Yuri A. Blinkov. Involutive bases of polynomial ideals. Mathematics and Computers in Simulation, 45:519–541, 1998.
5. 5.
Maurice Janet. Les systèmes d’équations aux dérivées partielles. Journal de Mathematique. 8e série, 3:65–151, 1920.Google Scholar
6. 6.
Jean-François Pommaret. Systems of Partial Differential Equations and Lie Pseudogroups, volume 14 of Mathematics and Its Applications. Gordon and Breach Science Publishers, Inc., One Park Avenue, New York, NY 10016, 1978.
7. 7.
Joseph Miller Thomas. Differential Systems. American Mathematical Society, New York, 1937.Google Scholar
8. 8.
A. Yu. Zharkov and Yuri A. Blinkov. Involution approach to solving systems of algebraic equations. In G. Jacob, N. E. Oussous, and S. Steinberg, editors, Proceedings of the 1993 International IMACS Symposium on Symbolic Computation, pages 11–16. IMACS, Laboratoire d’Informatique Fondamentale de Lille, France, 1993.Google Scholar