**Graduate Studies in Mathematics**

Volume: 3;
1994;
289 pp;
Hardcover

MSC: Primary 13;

Print ISBN: 978-0-8218-3804-4

Product Code: GSM/3

List Price: $48.00

AMS Member Price: $38.40

MAA member Price: $43.20

**Electronic ISBN: 978-1-4704-1139-8
Product Code: GSM/3.E**

List Price: $48.00

AMS Member Price: $38.40

MAA member Price: $43.20

# An Introduction to Gröbner Bases

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*William W. Adams; Philippe Loustaunau*

As the primary tool for doing explicit computations in polynomial rings in many variables, Gröbner bases are an important component of all computer algebra systems. They are also important in computational commutative algebra and algebraic geometry. This book provides a leisurely and fairly comprehensive introduction to Gröbner bases and their applications. Adams and Loustaunau cover the following topics: the theory and construction of Gröbner bases for polynomials with coefficients in a field, applications of Gröbner bases to computational problems involving rings of polynomials in many variables, a method for computing syzygy modules and Gröbner bases in modules, and the theory of Gröbner bases for polynomials with coefficients in rings. With over 120 worked out examples and 200 exercises, this book is aimed at advanced undergraduate and graduate students. It would be suitable as a supplement to a course in commutative algebra or as a textbook for a course in computer algebra or computational commutative algebra. This book would also be appropriate for students of computer science and engineering who have some acquaintance with modern algebra.

#### Readership

Advanced undergraduate and beginning graduate students in mathematics, computer science, applied mathematics, and engineering interested in computational algebra.

#### Reviews & Endorsements

The book is self-contained and does not assume an extensive knowledge of algebra. The style of the book is as elementary as the subject allows. All chapters are enriched by a large number of examples and exercises, the total number is over 120 worked-out examples and over 200 exercises. All these features make it an excellent textbook for a first course in the theory of Gröbner bases for advanced undergraduate or beginning graduate students. The reviewer also warmly recommends the book for independent study by students and researchers looking for a theoretical introduction to Gröbner bases.

-- Mathematical Reviews

The rich set of applications and exercises concentrates on pure higher algebra … The book is intended as a textbook for advanced undergraduates.

-- Zentralblatt MATH

Clearly written and has a great collection of exercises. It is the best textbook at this level … recommend it to colleagues.

-- O. M. G. Jenda, Auburn University

A very carefully crafted introduction to the theory and some of the applications of Gröbner bases … contains a wealth of illustrative examples and a wide variety of useful exercises, the discussion is everywhere well-motivated, and further developments and important issues are well sign-posted … has many solid virtues and is an ideal text for beginners in the subject … certainly an excellent text.

-- Bulletin of the London Mathematical Society

#### Table of Contents

# Table of Contents

## An Introduction to Grobner Bases

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- Chapter 1. Basic Theory of Gröbner Bases 116 free
- Chapter 2. Applications of Gröbner Bases 5368
- 2.1. Elementary Applications of Gröbner Bases 5368
- 2.2. Hilbert Nullstellensatz 6176
- 2.3. Elimination 6984
- 2.4. Polynomial Maps 7994
- 2.5. Some Applications to Algebraic Geometry 90105
- 2.6. Minimal Polynomials of Elements in Field Extensions 97112
- 2.7. The 3-Color Problem 102117
- 2.8. Integer Programming 105120

- Chapter 3. Modules and Gröbner Bases 113128
- 3.1. Modules 113128
- 3.2. Gröbner Bases and Syzygies 118133
- 3.3. Improvements on Buchberger's Algorithm 124139
- 3.4. Computation of the Syzygy Module 134149
- 3.5. Gröbner Bases for Modules 140155
- 3.6. Elementary Applications of Gröbner Bases for Modules 152167
- 3.7. Syzygies for Modules 161176
- 3.8. Applications of Syzygies 171186
- 3.9. Computation of Horn 183198
- 3.10. Free Resolutions 194209

- Chapter 4. Gröbner Bases over Rings 201216
- Appendix A. Computations and Algorithms 275290
- Appendix B. Well-ordering and Induction 277292
- References 279294
- List of Symbols 283298
- Index 285300
- Back Cover Back Cover1305