理想、簇与算法
Ideals, varieties, and algorithms
David A. Cox、David Cox、John Little、Donal O'Shea · 初版 1992 年
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代数几何是研究单变量或多变量多项式方程组的学科，探讨诸如以下问题：该系统是否有有限个解？如果是，该如何求得它们？如果存在无限个解，又该如何描述和处理它们？多项式方程组的解构成了一个称为variety的几何对象；与之对应的代数对象则是ideal。ideal与variety之间存在着密切联系，揭示了代数与几何之间的深层纽带。本书针对本科生水平撰写，
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你可能会喜欢 David Cox 写的《Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra》（理想、簇与算法：计算代数几何与交换代数导论）。
You might enjoy Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by David Cox.
……值得一读（而且在这个帖子里其他地方还没被提到过，比如 Axler 和 Pugh 就有人提过了）。Paolo Aluffi 的《Algebra: Chapter 0》、Paul Lockhart 的《Measurement》、Moore 与 Mertens 的《The Nature of Computation》、Cox、Little 和 O'Shea 的《Ideals, Varieties, and Algorithms》
…to be read (and have not been mentioned elsewhere in this thread, i.e. Axler and Pugh). Algebra: Chapter 0 by Paolo Aluffi Measurement by Paul Lockhart The Nature of Computation by Moore & Mertens Ideals, Varieties, and Algorithms by Cox, Little, & O'Shea
……以及代数几何（Algebraic Geometry）目前的现代视角，要较好地理解作为该领域最初动机的代数簇（algebraic varieties）其实并不难。Cox、Little 和 O'Shea 写的一本面向本科生的书《Ideals, Varieties, and Algorithms》在入门引导方面做得极好。书中还有一个非常酷的项目：通过计算正弦和余弦多项式方程的 Groebner bases（格罗布纳基）来定义位形……
…and current modern viewpoint of Algebraic Geometry, it's not hard to get a decent understanding of algebraic varieties which are the original motivation in the field. An undergrad book on the subject Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea does a great job of introducing the subject. And has a really cool project on calculating Groebner bases for polynomial equations in sin and cosine to define the configuration…
……http://en.wikipedia.org/wiki/Linear_equation_over_a_ring 要读懂它的背景需要花很长时间，我也没有足够的背景知识能把它提炼成我们想要的精确结果。注意这篇维基百科条目引用了这本书：Cox、Little 和 O'Shea 写的《Ideals, Varieties, and Algorithms》。这是该类主题中一本经典且通俗易懂的参考书。虽然这个话题很有意思，但我认为最好把它当成一项技术来看待。知道它的存在……
…http://en.wikipedia.org/wiki/Linear_equation_over_a_ring it would take a long time to read up the background to understand it, and I don't have the background to distill it to the exact result we want. Note the wiki article references this book: Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. This is a standard and approachable reference on this kind of topics. Although this topic is interesting, I think it's best to treating it as a technology. Knowing it's existence…
不过对于代数几何这块，Cox、Little 和 O'Shea 写的《Ideals, Varieties, and Algorithms》非常棒。
For the algebraic geometry side, though, *Ideals, Varieties and Algorithms" by Cox, Little and O'Shea is great.
如果你确实有一些数学背景，Karen Smith 等人写的《An Invitation to Algebraic Geometry》是一本短小精悍的好书，能直切核心思想。而 Cox、Little 和 O'Shea 的《Ideals, Varieties, and Algorithms》则是一本“计算几何的入门导引”。我觉得其中部分内容可能会非常吸引 HN 读者，它面向的读者群应该是数学专业的本科生。
If you do have some math background, "An Invitation to Algebraic Geometry" by Karen Smith et al is a lovely, slim volume that gets to important ideas right away. "Ideals, Varieties, and Algorithms" is an "invitation to computational geometry" by Cox, Little, and O'Shea. I think parts of it might really appeal to HN readers and it's supposed to be for an undergrad math major audience.