Passive Complete Orthonomic Systems of PDEs and Involutive Bases of Polynomial Modules

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


The objective of this article is to enlighten the relationship between the two classical theories of passive complete orthonomic systems of PDEs on the one hand and Gröbner bases of finitely generated modules over polynomial rings on the other hand. The link between both types of canonical forms are the involutive bases which are both, a particular type of Gröbner bases which carry some additional structure and a natural translation of the notion of passive complete orthonomic systems of linear PDEs with constant coefficients into the language of polynomial modules.

We will point out some desirable applications which a “good” notion of involutive bases could provide. Unfortunately, these desires turn out to collide and we will discuss the problem of finding a reasonable compromise.


Polynomial Ring Critical Pair Monomial Ideal Weyl Algebra Linear PDEs 
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.


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  1. W. W. Adams, P. Loustaunau. An Introduction to Gröbner Bases, Graduate Studies in Mathematics, Vol. 3, AMS Press, Providence, (1994).Google Scholar
  2. J. Apel. The Theory of Involutive Divisions and an Application to Hilbert Function Computations. J. Symb. Comp. 25, 683–704 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  3. J. Apel. Computational Ideal Theory in Finitely Generated Extension Rings. Theoretical Computer Science, 244/1–2, 1–33 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  4. J. Apel. On a Conjecture of R. P. Stanley; Part I — Monomial Ideals. MSRI Preprint 2001-004 (2001).Google Scholar
  5. J. Apel. On a Conjecture of R. P. Stanley; Part II — Quotients modulo Monomial Ideals. MSRI Preprint 2001-009 (2001).Google Scholar
  6. J. Apel. Stanley Decompositions and Riquier Bases. Talk presented at the Conference on Applications of Computer Algebra (ACA’01), Albuquerque (2001).Google Scholar
  7. J. Apel, R. Hemmecke. Detecting Unnecessary Reductions in an Involutive Basis Computation. Work in Progress.Google Scholar
  8. J. Apel, W. Laßner. An Extension of Buchberger’s Algorithm and Calculations in Enveloping Fields of Lie Algebras. J. Symb. Comp. 6, 361–370 (1988).CrossRefzbMATHGoogle Scholar
  9. D. Bayer, M. Stillman. A Criterion for Detecting m-Regularity. Invent. math. 87, 1–11 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  10. T. Becker, V. Weispfenning, in cooperation with H. Kredel. Gröbner Bases, A Computational Approach to Commutative Algebra. Springer, New York, Berlin, Heidelberg, (1993).zbMATHGoogle Scholar
  11. B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, University of Innsbruck, (1965).Google Scholar
  12. B. Buchberger. An Algorithmic Method in Polynomial Ideal Theory, Chapter 6 in: N.K. Bose, ed., Recent Trends in Multidimensional System Theory, D.Reidel Publ. Comp., (1985).Google Scholar
  13. D. Eisenbud. Commutative Algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, (1995).Google Scholar
  14. V. P. Gerdt, Yu. A. Blinkov. Involutive Bases of Polynomial Ideals. Mathematics and Computers in Simulation 45, 519–542 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  15. V. P. Gerdt, Yu. A. Blinkov, D. A. Yanovich. Fast Computation of Polynomial Janet Bases. Talk presented at the Conference on Applications of Computer Algebra (ACA’01), Albuquerque (2001).Google Scholar
  16. Giovini, A., Mora, T., Niesi, G., Robbiano, L., Traverso, C. (1991). “One sugar cube, please,” or Selection Strategies in the Buchberger Algorithm. In: Watt, S.M. (ed.), Proc. ISSAC’91, ACM Press, New York, pp. 49–54.Google Scholar
  17. R. Hemmecke. Dynamical Aspects of Involutive Bases Computations. SNSC’01, Linz (2001), This Volume.Google Scholar
  18. M. Janet. Lecons sur les systèmes d’equations aux dérivées partielles. Gauthier-Villars, Paris (1929).zbMATHGoogle Scholar
  19. D. Mall, On the Relation Between Gröbner and Pommaret Bases. AAECC 9/2, 117–123 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  20. H. M. Möller, T. Mora. New Constructive Methods in Classical Ideal Theory, J. Algebra 100, 138–178, (1986).zbMATHCrossRefMathSciNetGoogle Scholar
  21. J. F. Pommaret. Systems of Partial Differential Equations and Lie Pseudogroups. Gordan and Breach, New York (1978).zbMATHGoogle Scholar
  22. C. H. Riquier. Les systémes d’equations aux dérivées partielles. Gauthier-Villars, Paris (1910).Google Scholar
  23. R. P. Stanley. Linear Diophantine Equations and Local Cohomology. Invent. math. 68, 175–193 (1982).zbMATHCrossRefMathSciNetGoogle Scholar
  24. J. M. Thomas. Riquier’s Existence Theorems. Annals of Mathematics 30/2, 285–310 (1929).Google Scholar
  25. J. M. Thomas. Differential Systems. American Mathematical Society, New York, (1937).Google Scholar
  26. A. R. Tresse. Sur les invariants différentials des groupes continus de transformations. Acta Mathematica 18, 1–8 (1894).CrossRefMathSciNetGoogle Scholar
  27. A. Yu. Zharkov, Yu. A. Blinkov. Involution Approach to Solving Systems of Algebraic Equations. Proc. IMACS’93, 11–16 (1993).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Joachim Apel
    • 1
  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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