1. A Quick Introduction to Gröbner Bases

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.