An Introduction to Grobner Bases (Graduate Studies in Mathematics, Vol 3) (Graduate Studies in Mathematics, Vol 3)

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. 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 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. 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 As the primary tool for doing explicit computations in polynomial rings in many variables, Grobner bases are an important component of various computer algebra systems. They are also important in computational commutative algebra and algebraic geometry. This book offers a comprehensive introduction to Grobner bases and their applications. Chapter 1. Basic Theory Of Gröbner Bases Chapter 2. Applications Of Gröbner Bases Chapter 3. Modules And Gröbner Bases Chapter 4. Gröbner Bases Over Rings Appendix A. Computations And Algorithms Appendix B. Well-ordering And Induction William W. Adams, Philippe Loustaunau. Includes Bibliographical References And Index.