Gröbner Bases -

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Résumé :
The idea of the Gr?bner 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 Gr?bner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Gr?bner 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 Gr?bner basis. Since then, rapid development on the Gr?bner basis has been achieved by many researchers, including Bernd Sturmfels. This book serves as a standard bible of the Gr?bner 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 Gr?bner basis, requiring no special knowledge?to understand those basic points. Starting from the introductory performance of the Gr?bner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Gr?bner 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 Gr?bner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Gr?bner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Gr?bner 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.

Sommaire:

A Quick Introduction to Gr?bner Bases.- Warm-up Drills and Tips for Mathematical Software.- Computation of Gr?bner Bases.- Markov Bases and Designed Experiments.- Convex Polytopes and Gr?bner Bases.- Gr?bner Basis for Rings of Differential Operators and Applications.- Examples and Exercises.

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