Gröbner bases : statistics and software systems
著者
書誌事項
Gröbner bases : statistics and software systems
Springer, c2013
- タイトル別名
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Gröbner Dojo
グレブナー道場
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注記
Original edition published by Kyoritsu Shuppan Co., Ltd., Tokyo, 2011
Includes bibliographical references and index
内容説明・目次
内容説明
The idea of the Groebner 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 Groebner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Groebner 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 Groebner basis. Since then, rapid development on the Groebner basis has been achieved by many researchers, including Bernd Sturmfels.
This book serves as a standard bible of the Groebner 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 Groebner basis, requiring no special knowledge to understand those basic points.
Starting from the introductory performance of the Groebner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Groebner 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 Groebner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Groebner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Groebner 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.
目次
A Quick Introduction to Groebner Bases.- Warm-up Drills and Tips for Mathematical Software.- Computation of Groebner Bases.- Markov Bases and Designed Experiments.- Convex Polytopes and Groebner Bases.- Groebner Basis for Rings of Differential Operators and Applications.- Examples and Exercises.
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