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