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Gröbner Bases pp 223-278 | Cite as

Convex Polytopes and Gröbner Bases

  • Hidefumi Ohsugi
Chapter

Abstract

Gröbner bases of toric ideals have applications in many research areas. Among them, one of the most important topics is the correspondence to triangulations of convex polytopes. It is very interesting that, not only do Gröbner bases give triangulations, but also “good” Gröbner bases give “good” triangulations (unimodular triangulations). On the other hand, in order to use polytopes to study Gröbner bases of ideals of polynomial rings, we need the theory of Gröbner fans and state polytopes. The purpose of this chapter is to explain these topics in detail. First, we will explain convex polytopes, weight vectors, and monomial orders, all of which play a basic role in the rest of this chapter. Second, we will study the Gröbner fans of principal ideals, homogeneous ideals, and toric ideals; this will be useful when we analyze changes of Gröbner bases. Third, we will discuss the correspondence between the initial ideals of toric ideals and triangulations of convex polytopes, and the related ring-theoretic properties. Finally, we will consider the examples of configuration matrices that arise from finite graphs or contingency tables, and we will use them to verify the theory stated above. If you would like to pursue this topic beyond what is included in this chapter, we suggest the books [2, 7].

Keywords

Convex Polytopes Monomial Ideal Finite Graph Homogeneous Ideal Initial Ideal 
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.

References

  1. 1.
    W. Bruns, R. Hemmecke, B. Ichim, M. Köppe, C. Söger, Challenging computations of Hilbert bases of cones associated with algebraic statistics. Exp. Math. 20, 25–33 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    I.M. Gelfand, M.M. Kapranov, A.V. Zelevinski, Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications (Birkhauser, Boston, 1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    H. Ohsugi, T. Hibi, Normal polytopes arising from finite graphs. J. Algebra 207, 409–426 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    H. Ohsugi, T. Hibi, A normal (0,1)-polytope none of whose regular triangulations is unimodular. Discrete Comput. Geom. 21, 201–204 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    H. Ohsugi, T. Hibi, Toric ideals arising from contingency tables, in Commutative Algebra and Combinatorics, ed. by W. Bruns. Ramanujan Mathematical Society Lecture Notes Series, Number 4 (Ramanujan Mathematical Society, Mysore, 2007), pp. 91–115Google Scholar
  6. 6.
    H. Ohsugi, J. Herzog, T. Hibi, Combinatorial pure subrings. Osaka J. Math. 37, 745–757 (2000)zbMATHMathSciNetGoogle Scholar
  7. 7.
    M. Saito, B. Sturmfels, N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations. Algorithms and Computation in Mathematics, vol. 6 (Springer, Berlin, 2000)Google Scholar
  8. 8.
    A. Simis, W.V. Vasconcelos, R.H. Villarreal, The integral closure of subrings associated to graphs. J. Algebra 199, 281–289 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    B. Sturmfels, Gröbner Bases and Convex Polytopes (American Mathematical Society, Providence, 1996)zbMATHGoogle Scholar
  10. 10.
    S. Sullivant, Compressed polytopes and statistical disclosure limitation. Tohoku Math. J. 58, 433–445 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    R.R. Thomas, Lectures in Geometric Combinatorics. Student Mathematical Library, IAS/Park City Mathematical Subseries, vol. 33 (American Mathematical Society, Providence, 2006)Google Scholar
  12. 12.
    G.M. Ziegler, Lectures on Polytopes (Springer, New York, 1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of Mathematics, College of ScienceRikkyo UniversityTokyoJapan

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