Gröbner Bases pp 107-163 | Cite as

Computation of Gröbner Bases

  • Masayuki Noro


In Chap. 1, we presented the theoretical foundation of Gröbner bases, and many of our computations were carried out by hand. When we want to apply Gröbner bases to practical problems, however, in most cases, we will need the help of computers. There are many mathematical software systems which support the computation of Gröbner bases, but we will often encounter cases which require careful settings or preprocessing in order to be efficient. In this chapter, we explain various methods to efficiently use a computer to compute Gröbner bases. We also present some algorithms for performing operations on the ideals realized by Gröbner bases. These operations are implemented in several mathematical software systems: Singular, Macaulay2, CoCoA, and Risa/Asir. We will illustrate the usage of these systems mainly by example.


Polynomial Ring Hilbert Function Primary Decomposition Base Ring Initial Monomial 
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.


  1. 1.
    J. Abbott, A.M. Bigatti, CoCoALib: a C++ library for doing computations in commutative algebra.
  2. 2.
    CoCoA Team, A system for doing computations in commutative algebra.
  3. 3.
    W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 3-1-2—a computer algebra system for polynomial computations (2010).
  4. 4.
    R. Gebauer, H.M. Möller, On an installation of Buchberger’s algorithm. J. Symbolic Comput. 6, 275–286 (1988)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    A. Giovini, T. Mora, G. Niesi, L. Robbiano, C. Traverso, “One sugar cube, please” or selection strategies in the Buchberger algorithm, in Proceedings of the ISSAC 1991 (ACM, New York, 1991), pp. 49–54Google Scholar
  6. 6.
    D.R. Grayson, M.E. Stillman, Macaulay2, a software system for research in algebraic geometry.
  7. 7.
    T. Kawazoe, M. Noro, Algorithms for computing a primary ideal decomposition without producing intermediate redundant components. J. Symbolic Comput. 46, 1158–1172 (2011)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    M. Kreuzer, L. Robbiano, Computational Commutative Algebra 1 (Springer, Berlin, 2000); Computational Commutative Algebra 2 (Springer, Berlin, 2005)Google Scholar
  9. 9.
    M. Noro, N. Takayama, H. Nakayama, K. Nishiyama, K. Ohara, Risa/Asir: a computer algebra system.

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceKobe UniversityKobeJapan

Personalised recommendations