Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics)

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry―the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz―this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).

The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple™, Mathematica® and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.

From the reviews of previous editions:

 “…The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. …The book is well-written. …The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.”

 ―Peter Schenzel, zbMATH, 2007

 “I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.”

 ―The American Mathematical Monthly

本書涵蓋了代數幾何和交換代數的主題，並強調實際和計算方面。前四章是本書的核心。前言中的一個全面的圖表展示了在這些章節完成後繼續學習的多種方式。除了代數幾何的基礎知識 - 消去定理、擴展定理、閉包定理和零點定理 - 這個新版本還包含了幾個重大的改變，所有這些都在前言中列出。最大的修訂是增加了一個新的第十章，介紹了在過去幾十年中在計算格羅布納基礎方面取得的一些重要進展。本書還包括附錄C中的當前計算機代數材料和更新的獨立項目（附錄D）。

本書可以作為本科抽象代數的第一或第二門課程，並且可能需要一些補充材料，也可以作為開始研究生水平的代數幾何或計算代數課程。讀者的先備知識包括線性代數和以證明為導向的課程。假設讀者可以使用計算機代數系統。附錄C描述了Maple™、Mathematica®和Sage等與本文最相關的系統的功能。本文中使用了偽代碼；附錄B詳細描述了所使用的偽代碼。

從以前版本的評論中可以看出：

“...本書介紹了布赫伯格算法及其在syzygies、Hilbert多項式、主分解等方面的應用。它還介紹了傳統的代數幾何，並應用於理想成員問題、求解多項式方程和消去理論。...本書寫得很好。...評論者確信它將成為引導本科生進一步了解交換代數和代數幾何算法方面的優秀指南。” - Peter Schenzel, zbMATH, 2007

“我認為這本書非常出色。...說明非常清晰，有很多有用的圖片，還有很多有啟發性的練習，其中一些相當具有挑戰性...本書提供了現代交換代數和代數幾何的核心內容。” - The American Mathematical Monthly