Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics) 2nd Edition by David Cox (PDF)

Description

Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960’s. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing a new edition of Ideals, Varieties and Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout’s Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and REDUCE.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I used this text as a supplemental text for a course for one term. It was excellent as a supplement, but I was glad I was not using it as my main text for the course.This text is very clear and well-written, has tons of examples and develops geometric intuition for the subject quite well. Overall I think it is a great text for its target audience of undergraduate students with basically no background in abstract algebra. Unfortunately for a student who has had a course in abstract algebra the text goes too slowly, develops a lot of concepts they should already know in it and the explanations can be rather long at times.I highly recommend this text for its intended audience or as a supplement to a more advanced text for a reader who wants more examples, clearer statements, a more thorough development or a good source for geometric intuition about the subject.

⭐Wonderful find for used textbook. They delivered right on time.

⭐This book explains and illustrates the algorithms used by symbolic math packages such as Mathematica, Maple, CoCoA, MatLab, MuPAD,… to solve problems involving polynomials in many variables, and along the way teaches the elements of real algebraic geometry– most mathematics texts concentrate on the complex-variable version. It is not just for undergraduates; electrical engineers, for instance, should see it. Lots of pictures!

⭐All math books should be written like this.Just to echo prior reviews, the writing is clear, the pace perfect, and the voice is refreshingly non-condescending. There are so few math books which fit this bill I had to register my immense appreciation!The book is robust and comprehensive enough for self-study. It errs on the side of explaining surrounding ring theory as you go which obviates the need for wiki-ing every two minutes. In addition, the computational approach (through copious examples and exercises) gives you ready and frequent sanity checks on your understanding.Even if you simply want a foundation in “pure” algebraic geometry, this book will more than suffice. It does a remarkable job of motivating the journey into the more pure stuff, which even many pure books wouldn’t sully themselves with. If an equality or implication only goes one way, they’ll reliably provide counterexamples to demonstrate why. Hey, you may even find the detours into computers and robotics interesting (as did I)! As a follow-on text for a pure approach, try Perrin’s book.

⭐I don’t have the second edition of this book but did read the first, and the authors do a fine job of introducing the reader to the computational side of algebraic geometry. I will forego a chapter by chapter review therefore, but no doubt the second edition (which I do not own) is as well-written as the first. I would recommend it to anyone interested in the many applications of algebraic geometry and to those who need to understand how to compute things in algebraic geometry. The good thing about this book is that it gives a concrete flavor to a highly abstract subject. Algebraic geometry, through its applications to coding theory, cryptography, and computer graphics, is fast becoming the subject to learn. It is no longer just an esoteric, high-brow subject but one that is taking on major importance in the information age. Even without applications though it is a fascinating subject, and readers will get a taste of this in this book.

⭐This is the easiest introduction to algebraic geometry and commutative algebra, the authors had done a great job in writing a book that assume very little from the readers. To learn some algebraic geometry, you can either start with this book, or you can spend a year to read a lot of background materials in algebra and then go to a Graduate Text like Harris’ book. Of course, if you want to be an expert in algebra, you eventually need a lot of background, what this book can help you is to offer you a quick start, much quicker than you would ever imagine.

⭐excellent book. computational algebraic geometry. very useful.

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