Gröbner Bases

Statistics and Software Systems

Editors: Hibi, Takayuki (Ed.)

  • Covers broad aspects of Gröbner bases, including convex polytopes, algebraic statistics, and ring of differential operators
  • Discusses theoretical, practical, and computational aspects of Gröbner bases, providing information on how to use various software packages
  • Is readily accessible to graduate students, requiring no special knowledge to be understood
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  • ISBN 978-4-431-56215-3
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About this book

The idea of the Gröbner basis first appeared in a 1927 paper by F. S. Macaulay, who succeeded in creating a combinatorial characterization of the Hilbert functions of homogeneous ideals of the polynomial ring. Later, the modern definition of the Gröbner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Gröbner basis by Hironaka and Buchberger, it was not actively pursued for 20 years. A breakthrough was made in the mid-1980s by David Bayer and Michael Stillman, who created the Macaulay computer algebra system with the help of the Gröbner basis. Since then, rapid development on the Gröbner basis has been achieved by many researchers, including Bernd Sturmfels.

This book serves as a standard bible of the Gröbner basis, for which the harmony of theory, application, and computation are indispensable. It provides all the fundamentals for graduate students to learn the ABC’s of the Gröbner basis, requiring no special knowledge to understand those basic points.

Starting from the introductory performance of the Gröbner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Gröbner basis (Chapter 3). These three chapters may be regarded as the first act of a mathematical play. The second act opens with topics on algebraic statistics (Chapter 4), a fascinating research area where the Gröbner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Gröbner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Gröbner basis of the ring of differential operators gives effective algorithms on holonomic functions (Chapter 6). The third act (Chapter 7) is a collection of concrete examples and problems for Chapters 4, 5 and 6 emphasizing computation by using various software systems.

Table of contents (7 chapters)

  • A Quick Introduction to Gröbner Bases

    Hibi, Takayuki

    Pages 1-54

  • Warm-Up Drills and Tips for Mathematical Software

    Hamada, Tatsuyoshi

    Pages 55-106

  • Computation of Gröbner Bases

    Noro, Masayuki

    Pages 107-163

  • Markov Bases and Designed Experiments

    Aoki, Satoshi (et al.)

    Pages 165-221

  • Convex Polytopes and Gröbner Bases

    Ohsugi, Hidefumi

    Pages 223-278

Buy this book

eBook $99.00
price for USA (gross)
  • ISBN 978-4-431-54574-3
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $129.00
price for USA
  • ISBN 978-4-431-54573-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $129.00
price for USA
  • Customers within the U.S. and Canada please contact Customer Service at 1-800-777-4643, Latin America please contact us at +1-212-460-1500 (Weekdays 8:30am – 5:30pm ET) to place your order.
  • Due: October 13, 2016
  • ISBN 978-4-431-56215-3
  • Free shipping for individuals worldwide
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Gröbner Bases
Book Subtitle
Statistics and Software Systems
Editors
  • Takayuki Hibi
Copyright
2013
Publisher
Springer Japan
Copyright Holder
Springer Japan
eBook ISBN
978-4-431-54574-3
DOI
10.1007/978-4-431-54574-3
Hardcover ISBN
978-4-431-54573-6
Softcover ISBN
978-4-431-56215-3
Edition Number
1
Number of Pages
XV, 474
Number of Illustrations and Tables
123 b/w illustrations
Additional Information
Original Japanese edition published by Kyoritsu Shuppan Co., Ltd, Tokyo, 2011
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