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Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
This book details the heart and soul of modern commutative and algebraic geometry. It covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. 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 a significantly updated section on Maple; updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR; and presents a shorter proof of the Extension Theorem.
553 pages, Paperback
First published January 1, 1992
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About the author
David A. Cox
23 books7 followersDavid 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.
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.
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Displaying 1 - 4 of 4 reviews
July 26, 2021
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).
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).
January 31, 2023
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"
December 31, 2018
Boring and not useful.
May 29, 2025
I wish I had been miscarried
Displaying 1 - 4 of 4 reviews