Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra

by David A. Cox, Donal O'Shea, John Little

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.

Readers who are teaching from Ideals, Varieties, and Algorithms, or are studying the book on their own, may obtain a copy of the solutions manual by sending an email to jlittle@holycross.edu.

From the reviews of previous

 “…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

Genres: Mathematics, Geometry, Algebra, Computer Science, Reference, Logic

662 pages, Kindle Edition

First published January 1, 1992

About the author

David Archibald Cox (born September 23, 1948 in Washington, D.C.) is an American mathematician, working in algebraic geometry.

Cox graduated from Rice University with a Bachelor's degree in 1970 and his Ph.D. in 1975 at Princeton University, under the supervision of Eric Friedlander (Tubular Neighborhoods in the Etale Topology). From 1974 to 1975, he was assistant professor at Haverford College and at Rutgers University from 1975 to 1979. In 1979, he became assistant professor and in 1988 professor at Amherst College.

He studies, among other things, étale homotopy theory, elliptic surfaces, computer-based algebraic geometry (such as Gröbner basis), Torelli sets and toric varieties, and history of mathematics. He is also known for several textbooks. He is a fellow of the American Mathematical Society.

From 1987 to 1988 he was a guest professor at Oklahoma State University. In 2012, he received the Lester Randolph Ford Award for Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First.

Reviews

[Tinwerume · Rating 4 out of 5]

It's a weird book, but I like it. It's *very* concrete (as you might guess from the name). I wouldn't use it on its own because of that, but it's a good companion book for self study.

As an example of the weirdness: it talks about the Hilbert basis theorem purely in terms of generating sets of polynomials, and not Noetherian rings. The proof talks about ascending chain conditions, of course, but it doesn't really embed the theorem in the broader context of algebraic geometry (I have not finished the book, maybe it generalizes later).

Update after finishing: as you might expect from the name, it does a very good job of explaining the ideal - variety connection. This comes at the expense of not discussing e.g. the geometry of curves very much (compare the contents of this book to part 1 of A Royal Road to Algebraic Geometry, for instance).

[Siddharth · Rating 4 out of 5]

I quite enjoyed the low technology proofs of grobner bases and related facts. I wish some of the theory had been developed via more commutative algebra. I later found much more pleasing proofs in Ravi vakil's "the rising sea"

[Chris · Rating 2 out of 5]

Boring and not useful.

[Owen Levens · Rating 3 out of 5]

I wish I had been miscarried