An introduction to Gröbner bases.

*(English)*Zbl 0803.13015
Graduate Studies in Mathematics. 3. Providence, RI: American Mathematical Society (AMS). xiii, 289 p. (1994).

Chapters: 1. Basic theory of Gröbner bases, 2. Applications of Gröbner bases, 3. Modules and Gröbner bases, 4. Gröbner bases over rings.

The books begins on a very elementary level and introduces the polynomial arithmetic, the properties of Gröbner bases and Buchberger’s algorithm very carefully. The introduction is accompanied by several complete examples for the application of the algorithms. A significant part of the book is devoted to applications of the Gröbner bases. The book does not try to cover the complete field of computational ideal theory. Aspects like dimension theory, related algorithm methods, complexity, technology are redirected to different sources.

The rich set of applications and exercises concentrates on pure higher algebra. Especially the chapters on modules and bases over rings present material which is usually not available in that compact form. – The book is intended as a textbook for advanced undergraduates. It could have served also as a handbook for problems related to polynomial ideal algebra; however, the solutions of the numerous non-trivial problems are not included. The Gröbner base technique is handled on a pure theoretical level. Its limitations, especially the expression swell and the maximal sizes of practically computable problems are not mentioned.

The books begins on a very elementary level and introduces the polynomial arithmetic, the properties of Gröbner bases and Buchberger’s algorithm very carefully. The introduction is accompanied by several complete examples for the application of the algorithms. A significant part of the book is devoted to applications of the Gröbner bases. The book does not try to cover the complete field of computational ideal theory. Aspects like dimension theory, related algorithm methods, complexity, technology are redirected to different sources.

The rich set of applications and exercises concentrates on pure higher algebra. Especially the chapters on modules and bases over rings present material which is usually not available in that compact form. – The book is intended as a textbook for advanced undergraduates. It could have served also as a handbook for problems related to polynomial ideal algebra; however, the solutions of the numerous non-trivial problems are not included. The Gröbner base technique is handled on a pure theoretical level. Its limitations, especially the expression swell and the maximal sizes of practically computable problems are not mentioned.

Reviewer: H.Melenk (Berlin)

##### MSC:

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |