Gröbner Bases

Abstract

A large part of commutative algebra is formulated in nonconstructive ways. A typical example is Hilbert’s basis theorem (Corollary 2.13), which guarantees the existence of finite ideal bases without giving a method to construct them. But commutative algebra also has a large computational part, which has developed into a field of research of its own, called computational commutative algebra. This field has its own conferences, its own research community, and it has produced a considerable number of books within a short period of time. The goal of this part of the book is to give readers a glimpse into this rich field. To learn more, readers should consult any of the following books, which I list roughly chronologically: Becker and Weispfenning [3], Cox et al. [12 and 13], Adams and Loustaunau [1], Vasconcelos [51], Kreuzer and Robbiano [31 and 32], Greuel and Pfister [22], and Decker and Lossen [15]. Eisenbud’s book [17] also has a chapter on Gröbner bases.