Polynomial Rings and Gröbner Bases

Abstract

The purpose of Chapter 1 is to provide the reader with sufficient knowledge of the basic theory of Gröbner bases which is required for reading the later chapters. In Section 1.1, we study Dickson’s Lemma, which is a classical result in combinatorics. Gröbner bases are then introduced and Hilbert’s Basis Theorem and Macaulay’s Theorem follow. In Section 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 Buchberger’s criterion and Buchberger’s algorithm presented in Section 1.3. Furthermore, in Section 1.4, elimination theory will be introduced. This theory is very useful for solving a system of polynomial equations. Finally, in Section 1.5, we discuss the universal Gröbner basis of an ideal. This is a finite set of polynomials which is a Gröbner basis for the ideal with respect to any monomial order.