Gröbner Bases: Statistics and Software Systems

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.

Gröbner基礎的概念首次出現在F. S. Macaulay於1927年的一篇論文中，他成功地創造了對多項式環的齊次理想的Hilbert函數的組合特徵。之後，Gröbner基礎的現代定義由Heisuke Hironaka於1964年和Bruno Buchberger於1965年獨立引入。然而，在Hironaka和Buchberger發現Gröbner基礎的概念之後，它在接下來的20年內並未受到積極追求。在1980年代中期，David Bayer和Michael Stillman取得了突破，他們借助Gröbner基礎創建了Macaulay計算機代數系統。從那時起，包括Bernd Sturmfels在內的許多研究人員在Gröbner基礎上取得了快速發展。

這本書是Gröbner基礎的標準聖經，其中理論、應用和計算的和諧是不可或缺的。它為研究生提供了學習Gröbner基礎的ABC的所有基礎知識，不需要特殊知識來理解這些基本要點。

從Gröbner基礎的介紹性表演（第1章）開始，接著是對數學軟件的探討（第2章）。然後是對如何計算Gröbner基礎的深入討論（第3章）。這三章可以被視為數學劇的第一幕。第二幕以代數統計學的主題開始（第4章），這是一個迷人的研究領域，其中Gröbner基礎是馬爾可夫鏈蒙特卡羅方法的基本工具。此外，一個齊次理想的Gröbner基礎對凸多面體的研究產生了巨大影響（第5章）。此外，微分算子環的Gröbner基礎提供了關於保拓函數的有效算法（第6章）。第三幕（第7章）是對第4、5和6章的具體例子和問題的集合，強調使用各種軟件系統進行計算。