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Examples and Exercises

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Gröbner Bases


There are two aspects to the study of Gröbner bases: theory and computation. For problems which are difficult to solve by theoretical approaches, it may be possible to obtain solutions by computation, using either brute force or more elegant methods. On the other hand, for problems for which the computational methods are difficult, it may be possible to obtain solutions by a combination of theoretical insight and calculations. This is one of the attractions of Gröbner bases. Chapters 4–6 emphasized the theoretical aspect. In this chapter, we present problems and answers which utilize various software systems. It is our hope that readers will perform the calculations on these software systems while studying this chapter. Following these problems and their answers, we provide easy exercises which will help the reader to understand how to use these software systems to study or apply Gröbner bases. We will use computer algebra systems, statistical software systems, and some expert systems for polytopes and toric ideals; this covers several areas related to the theory and applications of Gröbner bases.

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  1. 1.

    Error messages may be displayed if Asir Contrib is not loaded since the package yang.rr uses some functions defined in Asir Contrib. If this happens, use the command import(‘‘names.rr’’);.

  2. 2.

    The command initialIdealW accomplishes this procedure in Singular. However, there may be a bug in version 3-1-2.


  1. F. Chyzak, An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Math. 217, 115–134 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Chyzak, Mgfun,

  3. J.A. De Loera et al., LattE.{}latte/

  4. W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 3-1-2 — A computer algebra system for polynomial computations.

  5. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. 1 (McGraw-Hill, 1953)

    Google Scholar 

  6. E. Gawrilow, M. Joswig, polymake: a framework for analyzing convex polytopes, in Polytopes — Combinatorics and Computation (2000), pp. 43–74.

  7. D.R. Grayson, M.E. Stillman, Macaulay2, a software system for research in algebraic geometry.

  8. R. Hemmecke, P.N. Malkin, Computing generating sets of lattice ideals and Markov bases of lattices. J. Symb. Comput. 44, 1463–1476 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. T. Hibi, Gröbner Bases (Asakura, Tokyo, 2003) (in Japanese)

    Google Scholar 

  10. Y. Iba, M. Tanemura, Y. Omori, H. Wago, S. Sato, A. Takahashi, Computational Statistics II. Frontiers of Statistical Science, vol. 12 (Iwanami, Tokyo, 2005) (in Japanese)

    Google Scholar 

  11. A.N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties.{}jensen/software/gfan/gfan.html

  12. Maple.

  13. M. Noro, An efficient modular algorithm for computing the global b-function, mathematical software, in Proceedings of ICMS2002 (World Scientific, Singapore, 2002), pp. 147–157

    Google Scholar 

  14. M. Noro, N. Takayama, Risa/Asir drill. (2012, in Japanese)

  15. M. Noro et al., Risa/Asir.

  16. T. Oaku, Computation of the characteristic variety and the singular locus of a system. Jpn. J. Ind. Appl. Math. 11, 485–497 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Oaku, D-modules and Computational Mathematics (Asakura, Tokyo, 2002) (in Japanese)

    Google Scholar 

  18. R Development Core Team, R: A language and environment for statistical computing.

  19. J. Rambau, TOPCOM: triangulations of point configurations and oriented matroids, in Mathematical Software—ICMS 2002 (2002), pp. 330–340.

  20. M. Saito, B. Sturmfels, N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations (Springer, Berlin, 2000)

    Book  MATH  Google Scholar 

  21. A. Schrijver, Theory of Linear and Integer Programming (Wiley Interscience, New York, 1986)

    MATH  Google Scholar 

  22. B. Sturmfels, Gröbner Bases and Convex Polytopes. University lecture series 8 (American Mathematical Society, Providence, 1995)

    Google Scholar 

  23. N. Takayama, Kan/sm1.

  24. Wolfram MathWorld, Horn Function.

  25. G. M. Ziegler, Lectures on Polytopes (Springer, Berlin, 1995)

    Book  MATH  Google Scholar 

  26. 4ti2 team, 4ti2—a software package for algebraic, geometric and combinatorial problems on linear spaces.

Download references

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Correspondence to Hiromasa Nakayama .

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Nakayama, H., Nishiyama, K. (2013). Examples and Exercises. In: Hibi, T. (eds) Gröbner Bases. Springer, Tokyo.

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