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Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics) Hardcover – July 31, 2008

ISBN-13: 978-0387356501 ISBN-10: 0387356509 Edition: 3rd

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Product Details

  • Series: Undergraduate Texts in Mathematics
  • Hardcover: 553 pages
  • Publisher: Springer; 3rd edition (July 31, 2008)
  • Language: English
  • ISBN-10: 0387356509
  • ISBN-13: 978-0387356501
  • Product Dimensions: 6.1 x 1.2 x 9.2 inches
  • Shipping Weight: 2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.8 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Best Sellers Rank: #530,694 in Books (See Top 100 in Books)

Editorial Reviews

Review

From the reviews of the third edition:

"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 a excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry." (Peter Schenzel, Zentralblatt MATH, Vol. 1118 (20), 2007)

From the Back Cover

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. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.

In addition to enhancing the text of the second edition, with over 200 pages reflecting changes to enhance clarity and correctness, this third edition of Ideals, Varieties and Algorithms includes:

A significantly updated section on Maple in Appendix C

Updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR

A shorter proof of the Extension Theorem presented in Section 6 of Chapter 3

From the 2nd Edition:

"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


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Most Helpful Customer Reviews

13 of 15 people found the following review helpful By Abstract Space on February 10, 2008
Format: Hardcover
This is the 3rd edition of a popular reference on the subject. There was some production error with earlier version of the book. Even with the latest version, the authors have provided 14-page worth of corrections for the 1st printing of the 3rd edition. See [...]
So buyers may be better off waiting for a corrected later version from the publisher.
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8 of 9 people found the following review helpful By CompEngGradStudent on February 11, 2009
Format: Hardcover
This book explains the concepts well and the proofs are written carefully and clearly. This is a math book, so you should expect the usual Definition-Theorem-Proof format, but the exposition is progressive, and the material is very well motivated. I had to implement an algorithm for the computation of Groebner bases, and I borrowed this book from the University library to learn about the Buchberger algorithm and related topics, and thanks to this book I was able to understand this algorithm and its correctness proof without the help of anyone. Of course Buchberger's algorithm is only a small part of this book, and you can find a lot of important topics in it.
Finally, note that the second part of Madhu Sudan's Algebra and Computation course at MIT follows this book closely. So at least one top expert in the field seems to have a good opinion of the book.
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1 of 1 people found the following review helpful By Danny on April 23, 2013
Format: Hardcover Verified Purchase
Hello. I got this book as a student and it reads fairly well.

You need some algebra background although the book inserts some quick algebra in there. Example it explains what rings are, fields, ideals, etc. Basic ring theory. So dont expect anything too fancy.

After that, the book introduces the topic with every now and then backtracks to algebra for review. Example, it presents ideal properties, radical ideal properties before it introduces a definition from alg geo that includes them in there.

Essesntial, you work with fields, and polynomial rings, and affine space. So expect all the proofs to be focused on that, even though they may be true for any ring, not just polynomial rings. True for some not all.

Other than that, it is a good read, nothing fancy, pretty concrete and for self learning its pretty good. Check it out.
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By M-Nugget on September 8, 2013
Format: Hardcover Verified Purchase
I recently completed my honour's degree in mathematics, my research subject was Groebner bases, this text was my bible throughout the process
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