Introduction to Gröbner Bases

Summary

Outline

A comprehensive treatment of Gröbner bases theory is far beyond what can be done in one article in a book. Recent text books on Gröbner bases like (Becker, Weispfenning 1993) and (Cox, Little, O'Shea 1992) present the material on several hundred pages. However, there are only a few key ideas behind Gröbner bases theory. It is the objective of this introduction to explain these ideas as simply as possible and to give an overview of the immediate applications. More advanced applications are described in the other tutorial articles in this book.

The concept of Gröbner bases together with the characterization theorem (by “S-polynomials”) on which an algorithm for constructing Gröbner bases hinges has been introduced in the author's PhD thesis (Buchberger 1965), see also the journal publication (Buchberger 1970). In these early papers we also gave some first applications (computation in residue class rings modulo polynomial ideal congruence, algebraic equations, and Hilbert function computation), a computer implementation (in the assembler language of the ZUSE Z23V computer), and some first remarks on complexity. Later work by the author and by many other authors has mainly added generalizations of the method and quite a few new applications for the algorithmic solution of various fundamental problems in algebraic geometry (polynomial ideal theory, commutative algebra).