Grbner Bases

The idea of the Grbner 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 Grbner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Grbner 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 Grbner basis. Since then, rapid development on the Grbner basis has been achieved by many researchers, including Bernd Sturmfels. This book serves as a standard bible of the Grbner basis, for which the harmony of theory, application, and computation are indispensable. It provides all the fundamentals for graduate students to learn the ABCs of the Grbner basis, requiring no special knowledgeto understand those basic points. Starting from the introductory performance of the Grbner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Grbner 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 Grbner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Grbner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Grbner 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 softwaresystems.