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An Introduction to Gröbner Bases
William W. Adams, University of Maryland, College Park, MD, and Philippe Loustaunau, George Mason University, Fairfax, VA
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Graduate Studies in Mathematics
1994; 289 pp; hardcover
Volume: 3
Reprint/Revision History:
reprinted with corrections 1996; third printing 2000
ISBN-10: 0-8218-3804-0
ISBN-13: 978-0-8218-3804-4
List Price: US$45
Member Price: US$36
Order Code: GSM/3
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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.

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Readership

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

Reviews

"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

  • Basic theory of Gröbner bases
  • Applications of Gröbner bases
  • Modules and Gröbner bases
  • Gröbner bases over rings
  • Appendix A. Computations and algorithms
  • Appendix B. Well-ordering and induction
  • References
  • List of symbols
  • Index
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