Comprehensive Gröbner Bases in a Java Computer Algebra System

  • Heinz Kredel
Conference paper


We present an implementation of the algorithms for computing comprehensive Gröbner bases in a Java computer algebra system (JAS). Contrary to approaches to implement comprehensive Gröbner bases with minimal requirements to the computer algebra system, we aim to provide all necessary algebraic structures occurring in the algorithm. In the implementation of a condition we aim at the maximal semantic exploitation of the occurring algebraic structures: the set of equations that equal zero are implemented as an ideal (with Gröbner base computation) and the set of inequalities are implemented as a multiplicative set which is simplified to polynomials of minimal degrees using squarefree or irreducible decomposition. The performance of our implementation is compared on well-known examples. With our approach we can also make the transition of a comprehensive Gröbner system to a polynomial ring over a regular coefficient ring and test or compute Gröbner bases.



I thank Thomas Becker for discussions on the implementation of a polynomial template library and Raphael Jolly for the discussions on the generic type system suitable for a computer algebra system. JAS itself was improved by requirements from various users, especially Axel Kramer and by valuable feedback from other colleagues, in particular by Dongming Wang, Thomas Sturm, and Wolfgang K. Seiler, to name a few. This paper profited moreover from comments and feedback we received at the conference. Thanks also to Markus Aleksy and Hans-Günther Kruse for encouraging and supporting this work.


  1. 1.
    Montes, A.: An new algorithm for discussing Gröbner basis with parameters. J. Symb. Comput. 33(1–2), 183–208 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Suzuki, A., Sato, Y.: An alternative approach to comprehensive Gröbner bases. J. Symb. Comput. 36(3–4), 649–667 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Weispfenning, V.: Comprehensive Gröbner bases. J. Symb. Comp. 14(1), 1–29 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Weispfenning, V.: Canonical comprehensive Gröbner bases. In: ISSAC 2002, pp. 270–276. ACM (2002)Google Scholar
  5. 5.
    Kredel, H.: On the design of a Java computer algebra system. In: Proceedings of PPPJ 2006, pp. 143–152. University of Mannheim (2006)Google Scholar
  6. 6.
    Kredel, H.: Evaluation of a Java computer algebra system. In: Proceedings ASCM 2007, pp. 59–62. National University of Singapore (2007)Google Scholar
  7. 7.
    Kredel, H.: Evaluation of a Java computer algebra system. Lect. Notes Artif. Intell. 5081, 121–138 (2008)Google Scholar
  8. 8.
    Kredel, H.: On a Java computer algebra system, its performance and applications. Sci. Comput. Program. 70(2–3), 185–207 (2008)Google Scholar
  9. 9.
    Kredel, H.: Comprehensive Gröbner bases in a Java computer algebra system. In: Proceedings ASCM 2009, pp. 77–90. Kyushu University, Fukuoka, Japan (2009)Google Scholar
  10. 10.
    Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In: Proceedings of ISSAC 2006, pp. 326–331 (2006)Google Scholar
  11. 11.
    Dolzmann, A., Sturm, T.: Redlog: computer algebra meets computer logic. ACM SIGSAM Bull. 31(2), 2–9 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kredel, H., Pesch, M.: MAS: The Modula-2 Algebra System. Computer Algebra Handbook, pp. 421–428. Springer, Heidelberg (2003)Google Scholar
  13. 13.
    Schönfeld, E.: Parametrische Gröbnerbasen im Computer Algebra System ALDES / SAC-2. Diplomarbeit. Universität Passau, Passau (1991)Google Scholar
  14. 14.
    Weispfenning, V.: Gröbner bases for polynomial ideals over commutative regular rings. In: Davenport, H. (ed.) Proceedings of ISSAC’87, pp. 336–347. Springer (1987)Google Scholar
  15. 15.
    Weispfenning, V.: Comprehensive Gröbner bases and regular rings. J. Symb. Comput. 41, 285–296 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Montes, A., Manubens, M.: Improving DISPGB algorithm using the discriminant ideal. J. Symb. Comput. 41, 1245–1263 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kapur, D.: An approach for solving systems of parametric polynomial equations. Principles and Practices of Constraint Programming, pp. 217–244. MIT Press, Cambridge (1995)Google Scholar
  18. 18.
    Inoue, S., Sato, Y.: On the parallel computation of comprehensive Gröbner systems. In: Proceedings of PASCO’07, pp. 99–101 (2007)Google Scholar
  19. 19.
    Nabeshima, K.: A speed-up of the algorithm for computing comprehensive Gröbner systems. In: Proceedings of ISSAC 2007, pp. 299–306 (2007)Google Scholar
  20. 20.
    Nabeshima, K.: PGB: A package for computing parametric polynomial systems. In: Proceedings ASCM 2009, pp. 111–122. Kyushu University, Fukuoka, Japan (2009)Google Scholar
  21. 21.
    Kredel, H.: The Java algebra system (JAS). Technical report, (2000)
  22. 22.
    Kredel, H.: Multivariate greatest common divisors in the Java computer algebra system. In: Proceedings of Automated Deduction in Geometry (ADG), pp. 41–61. East China Normal University, Shanghai (2008)Google Scholar
  23. 23.
    Jolly, R., Kredel, H.: Generic, type-safe and object oriented computer algebra software. In: Proceedings of CASC 2010, pp. 162–177. Springer LNCS 6244 (2010)Google Scholar
  24. 24.
    Gräbe, H.G.: The Symbolicdata project. Technical report, (2000–2006) Accessed March 2010
  25. 25.
    Sato, Y., Suzuki, A.: Gröbner bases in polynomial rings over von Neumann regular rings - their applications. In: Proceedings ASCM 2000, World Scientific Publications, Lecture Notes Series on Computing, vol. 8, pp. 59–62 (2000)Google Scholar
  26. 26.
    Sato, Y., Nagai, A., Inoue, S.: On the computation of elimination ideals of boolean polynomial rings. In: ASCM, pp. 334–348 (2007)Google Scholar
  27. 27.
    Kredel, H.: Solvable Polynomial Rings. Dissertation, Universität Passau (1993)Google Scholar
  28. 28.
    Kredel, H.: Distributed parallel Gröbner bases computation. In: Proceedings of Workshop on Engineering Complex Distributed Systems at CISIS 2009, CD-ROM. Fukuoka Insitute of Technology, Japan (2009)Google Scholar
  29. 29.
    Kredel, H.: Distributed hybrid Gröbner bases computation. In: Proceedings of Workshop on Engineering Complex Distributed Systems at CISIS 2010. CD-ROM. University of Krakow, Poland (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.IT-CenterUniversity of MannheimMannheimGermany

Personalised recommendations