Abstract
Neither specialist knowledge nor extensive investment of time is required in order for a nonspecialist to learn fundamentals on Gröbner bases. The purpose of this chapter is to provide the reader with sufficient understanding of the theory of Gröbner bases as quickly as possible with the assumption of only a minimum of background knowledge. In Sect. 1.1, the story starts with Dickson’s Lemma, which is a classical result in combinatorics. The Gröbner basis is then introduced and Hilbert Basis Theorem follows. With considering the reader who is unfamiliar with the polynomial ring, an elementary theory of ideals of the polynomial ring is also reviewed. In Sect. 1.2, the division algorithm, which is the framework of Gröbner bases, is discussed with a focus on the importance of the remainder when performing division. The highlights of the fundamental theory of Gröbner bases are, without doubt, Buchberger criterion and Buchberger algorithm. In Sect. 1.3 the groundwork of these two items are studied. Now, to read Sects. 1.1–1.3 is indispensable for being a user of Gröbner bases. Furthermore, in Sect. 1.4, the elimination theory, which is effective technique for solving simultaneous equations, is discussed. The toric ideal introduced in Sect. 1.5 is a powerful weapon for the application of Gröbner bases to combinatorics on convex polytopes. Clearly, without toric ideals, the results of Chaps. 4 and 4 could not exist. The Hilbert function studied in Sect. 1.6 is the most fundamental tool for developing computational commutative algebra and computational algebraic geometry. Section 1.6 supplies the reader with sufficient preliminary knowledge to read Chaps. 5 and 6. However, since the basic knowledge of linear algebra is required for reading Sect. 1.6, the reader who is unfamiliar with linear algebra may wish to skip Sect. 1.6 in his/her first reading. Finally, in Sect. 1.7, the historical background of Gröbner bases is surveyed with providing references for further study.
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Notes
- 1.
A monomial order is also called a term order(ing).
- 2.
Some authors prefer ≺ to < .
- 3.
A reverse lexicographic order is also called a graded reverse lexicographic order(ing).
References
W.W. Adams, P. Loustaunau, An Introduction to Gröbner Bases (American Mathematical Society, Providence, 1994)
S. Aoki, T. Hibi, H. Ohsugi, A. Takemura, Gröbner bases of nested configurations. J. Algebra 320, 2583–2593 (2008)
S. Aoki, T. Hibi, H. Ohsugi, A. Takemura, Markov basis and Gröbner basis of Segre–Veronese configuration for testing independence in group-wise selections. Ann. Inst. Stat. Math. 62, 299–321 (2010)
T. Becker, W. Weispfenning, Gröbner Bases (Springer, New York, 1993)
B. Buchberger, An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, Ph.D. Dissertation (University of Innsbruck, 1965)
P. Conti, C. Traverso, Buchberger algorithm and integer progamming, in Applied Algebra, Algebraic Algorithms and Error Correcting Codes, ed. by H. Mattson, T. Mora, T. Rao. Lecture Notes in Computer Science, vol. 539 (Springer, Berlin, 1991), pp. 130–139
D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms (Springer, Berlin, 1992)
D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry. Graduate Texts in Mathematics, vol. 185 (Springer, Berlin, 1998)
P. Diaconis, B. Sturmfels, Algebraic algorithms for sampling from conditional distributions. Ann. Stat. 26, 363–397 (1998)
D. Eisenbud, Commutative Algebra with a View Towards Algebraic Geometry. Graduate Texts in Mathematics, vol. 150 (Springer, Berlin, 1995)
I.M. Gel’fand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994)
J. Herzog, T. Hibi, Monomial Ideals. Graduate Texts in Mathematics, vol. 260 (Springer, Berlin, 2010)
T. Hibi, Algebraic Combinatorics on Convex Polytopes (Carslaw Publications, Glebe, 1992)
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79, 109–203, 205–326 (1964)
F.S. Macaulay, Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc. 26, 531–555 (1927)
T. Oaku, An algorithm of computing b-functions. Duke Math. J. 87, 115–132 (1997)
T. Oaku, Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules. Adv. Appl. Math. 19, 61–105 (1997)
H. Ohsugi, T. Hibi, A normal (0, 1)-polytope none of whose regular triangulations is unimodular. Discrete Comput. Geom. 21, 201–204 (1999)
H. Ohsugi, T. Hibi, Toric ideals generated by quadratic binomials. J. Algebra 218, 509–527 (1999)
H. Ohsugi, T. Hibi, Compressed polytopes, initial ideals and complete multipartite graphs. Ill. J. Math. 44, 391–406 (2000)
H. Ohsugi, T. Hibi, Toric ideals arising from contingency tables, in Commutative Algebra and Combinatorics. Ramanujan Mathematical Society Lecture Notes Series, vol. 4 (Ramanujan Mathematical Society, Mysore, 2007), pp. 91–115
J.-E. Roos, B. Sturmfels, A toric ring with irrational Poincaré–Betti series. C. R. Acad. Sci. Paris Ser. I Math. 326, 141–146 (1998)
M. Saito, B. Sturmfels, N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations (Springer, Berlin, 2000)
R.P. Stanley, The upper bound conjecture and Cohen–Macaulay rings. Stud. Appl. Math. 54, 135–142 (1975)
R.P. Stanley, Combinatorics and Commutative Algebra, 2nd edn. (Birkhäuser, Boston, 1996)
B. Sturmfels, Gröbner Bases and Convex Polytopes (American Mathematical Society, Providence, 1996)
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Hibi, T. (2013). A Quick Introduction to Gröbner Bases. In: Hibi, T. (eds) Gröbner Bases. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54574-3_1
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