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

by David A. Cox, John B. Little

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.

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

551 pages, Paperback

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