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Three Remarks on Comprehensive Gröbner and SAGBI Bases

  • Manfred G-bel
  • Patrick Maier
Conference paper

Abstract

This note presents new complexity results for the comprehensive Gröbner bases (CGB) algorithm in the special case of one main variable and two polynomials, a general remark about CGB for parameterized binomial ideals, and it introduces the concept of comprehensive SAGBI bases together with a first application in invariant theory. Keywords. Comprehensive Gröbner bases, parameterized binomial ideals, comprehensive SAGBI bases, algorithmic invariant theory, permutation groups.

Keywords

Symmetric Group Minimal Degree Permutation Group Reduction Step Symbolic Computation 
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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References

  1. 1.
    Attardi, G., Traverso, C. (1996). Strategy-Accurate Parallel Buchberger Algorithms. Journal of Symbolic Computation 21(4–6), 411–426MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bigatti, A. M., La Scala, R., Robbiano, L. (1999). Computing Toric Ideals. Journal of Symbolic Computation 27(4),351–365MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Collart, S., Kalkbrener, M., Mall, D. (1997). Converting Bases with the Gröbner Walk. Journal of Symbolic Computation 24(3-4), 465–470MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Conti, P., Traverso, C. (1991). Buchberger Algorithm and Integer Programming. In: Mattson, H. F., Mora, T., Rao, T. R. N., (eds.), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 6th IntI. Conf., AAECC-91, volume 539 of LNCS, Springer, 130–139Google Scholar
  5. 5.
    Czapor, S. R. (1989). Solving Algebraic Equations: Combining Buchberger’s Algorithm with Multivariate Factorization. Journal of Symbolic Computation 7, 49–53 6._Dolzmann, A., Sturm, T. (1999). Redlog User Manual. MIP-9905, Fakultät für Mathematik und Informatik, Universität Passau.Google Scholar
  6. 7.
    Eisenbud, D. (1994). Commutative Algebra with a View Towards Algebraic Geometry. SpringerGoogle Scholar
  7. 8.
    Eisenbud, D., Sturmfels, B. (1996). Binomial Ideals. Duke Mathematical Journal 84, 1–45.MathSciNetMATHCrossRefGoogle Scholar
  8. 9.
    Gatermann, K. (1990). Symbolic Solution of Polynomial Equation Systems with Symmetry. In: Watanabe, Sh., Nagata, M., (eds.), International Symposium on Symbolic and Algebraic Computation ISSAC’90, 112–119, Tokyo, Japan, August, ACM PGoogle Scholar
  9. 10.
    Giovanni, 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.), International Symposium on Symbolic and Algebraic Computation, ISSAC- 91, ACM, 49–54Google Scholar
  10. 11.
    Göbel, M. (1995). Computing Bases for Permutation-Invariant Polynomials. Journal of Symbolic Computation 19, 285–291MathSciNetMATHCrossRefGoogle Scholar
  11. 12.
    Göbel, M. (1998). A Constructive Description of SAGBI Bases for Polynomial Invariants of Permutation Groups. Journal of Symbolic Computation 26, 261–272MathSciNetMATHCrossRefGoogle Scholar
  12. 13.
    Kapur, D., Madlener, K. (1989). A Completion Procedure for Computing a Canonical Basis of a k-Subalgebra. In: Kaltofen, E., Watt, S., (eds.), Proceedings of Computers and Mathematics 89. MIT, Cambridge, 1–11Google Scholar
  13. 14.
    Khuller, S., Göbel, M., Walter, J. (1999). Bases for Polynomial Invariants of Conjugates of Permutation Groups. Journal of Algorithms 32(1), 58–61MathSciNetCrossRefGoogle Scholar
  14. 15.
    Koppenhagen, U., Mayr, E. W. (1999). An Optimal Algorithm for Constructing the Reduced Gröbner Basis of Binomial Ideals. Journal of Symbolic Computation 28(3), 317–338MathSciNetMATHCrossRefGoogle Scholar
  15. 16.
    Kredel, H. (1990). MAS: Modula-2 Algebra System. In: Gerdt, V. P., Rostovtsev, V. A., and Shirkov, D. V. (eds.), IV International Conference on Computer Algebra in Physical Research. World Scientific Publishing Co., Singapore, 31–34Google Scholar
  16. 17.
    Moller, H. M., Mora, F. (1984). Upper and Lower Bounds for the Degree of Gröbner Bases. In: Fitch, J., (ed.), International Symposium on Symbolic and Algebraic Computation, EUROSAM-84, volume 174 of LNCS, Springer, 172–183Google Scholar
  17. 18.
    Mora, T. (1994). An Introduction to Commutative and Non-Commutative Gröbner Bases. Theoretical Computer Science 134(1), 131–173MathSciNetMATHCrossRefGoogle Scholar
  18. 19.
    Robbiano, L., Sweedler, M. (1990). Subalgebra Bases. In: Bruns, W., Simis, A., (eds.), Commutative Algebra (Lect. Notes Math. 1430). Springer, 61–87Google Scholar
  19. 20.
    Schonfeld, E. (1991). Parametrische Gröbner Basen im Computeragebra System ALDES/SAC-2, Diplomarbeit, Universität Passau.Google Scholar
  20. 21.
    Sturmfels, B. (1993). Algorithms in Invariant Theory. SpringerGoogle Scholar
  21. 22.
    Sturmfels, B. (1995). Gröbner Bases and Convex Polytopes. AMS University Lecture Series, Vol. 8, Providence RIGoogle Scholar
  22. 23.
    Traverso, C. (1996). Hilbert Functions and the Buchberger Algorithm. Journal of Symbolic Computation 22(4), 355–376MathSciNetMATHCrossRefGoogle Scholar
  23. 24.
    Weispfenning, V. (1987). Some Bounds for the Construction of Grobner Bases. In: Beth, T., Clausen, M., (eds.), Applicable Algebra, Error-Correcting Codes, Combinatorics and Computer Algebra, 4th Inti. Conf., AAECC-87, volume 307 of LNCS, Springer, 195–201Google Scholar
  24. 25.
    Weispfenning, V. (1989). Constructing Universal Gröbner Bases. In: Huguet, L., Poli, A., (eds.), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 5th IntI. Conf., AAECC-89, volume 356 of LNCS, Springer, 195–201Google Scholar
  25. 26.
    Weispfenning, V. (1992). Comprehensive Gröbner Bases. Journal of Symbolic Computation 14(1), 1–30MathSciNetMATHCrossRefGoogle Scholar
  26. 27.
    Zimmer, H.-G. (1998). SIMATH-Manual. Fachbereich Mathematik, Universtät Saarbrücken / Siemens AG München. (http://emmy.math.uni-sb.deisimath/)

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Manfred G-bel
    • 1
  • Patrick Maier
    • 2
  1. 1.Department of Electronic Systems EngineeringUniversity of EssexColchesterUK
  2. 2.Max-Planck-Institut für 1nformatikSaarbrückenGermany

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