An Introduction to Grobner Bases

✍ Scribed by Philippe Loustaunau William W. Adams

• Publisher: American Mathematical Society
• Year: 1994
• Tongue: English
• Leaves: 306
• Series: Graduate Studies in Mathematics, Vol 3
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover

An Introduction to Gröbner Bases

Copyright

 1994 by the American Mathematical Society

 ISBN 0-8218-3804-0

 QA251.3.A32 1994 512' .4--dc20

 LCCN 94-19081

Dedication

Contents

Preface

Chapter 1. Basic Theory of Grabner Bases

 1.1. Introduction.

 1.2. The Linear Case.

 1.3. The One Variable Case

 1.4. Term Orders

 1.5. Division Algorithm

 1.6. Grobner Bases

 1.7. S-Polynomials and Buclaberger's Algorithm

 1.8. Reduced Grobner Bases.

 1.9. Summary

Chapter 2. Applications of Grobner Bases

 2.1. Elementary Applications of Grobner Bases

 2.2. Hilbert Nullstellensatz

 2.3. Elimination.

 2.4. Polynomial Maps.

 2.5. Some Applications to Algebraic Geometry.

 2.6. Minimal Polynomials of Elements in Field Extensions

 2.7. The 3-Color Problem.

 2.8. Integer Programming.

Chapter 3. Modules and Grobner Bases

 3.1. Modules

 3.2. Grobner Bases and Syzygies.

 3.3. Improvements on Buchberger's Algorithm.

 3.4. Computation of the Syzygy Module.

 3.5. Grobner Bases for Modules

 3.6. Elementary Applications of Grobner Bases for Modules

 3.7. Syzygies for Modules

 3.8. Applications of Syzygies

 3.9. Computation of Hom.

 3.10. Free Resolutions

Chapter 4. Grobner Bases over Rings

 4.1. Basic Definitions

 4.2. Computing Grobner Bases over Rings.

 4.3. Applications of Grobner Bases over Rings

 4.4. A Prinxality Test.

 4.5. Grobner Bases over Principal Ideal Domains.

 4.6. Primary Decomposition in R[x] for R a PID

Appendix A. Computations and Algorithms

 Computation

 Algorithms

Appendix B. Well-ordering and Induction

References

List of Symbols

Index

Back Cover