Ideals, Varieties, and Algorithms
An Introduction to Computational Algebraic Geometry and Commutative Algebra
Authors: Cox, David A, Little, John, Oshea, Donal
Free Preview New edition extensively revised and updated
 Covers important topics such as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry and dimension theory
 Fourth edition includes updates on the computer algebra and independent projects appendices
 Features new central theoretical results such as the elimination theorem, the extension theorem, the closure theorem and the nullstellensatz
 Discusses some of the newer approaches to computing Groebner bases
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 About this Textbook

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 prooforiented 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.
From the reviews of previous editions:
“…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 wellwritten. …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
 About the authors

David A. Cox is currently Professor of Mathematics at Amherst College. John Little is currently Professor of Mathematics at College of the Holy Cross. Donal O'Shea is currently President and Professor of Mathematics at New College of Florida.
 Reviews

“In each of the new editions the authors' were interested to incorporate new developments, simplifications of arguments as well as further applications. Thanks to the authors' this is also the case in the present fourth edition. … Thanks to the continuously updating the textbook will remain an excellent source for the computational Commutative Algebra for students as well as for researchers interested in learning the subject.” (Peter Schenzel, zbMATH 1335.13001, 2016)
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 Table of contents (11 chapters)


Geometry, Algebra, and Algorithms
Pages 147

Gröbner Bases
Pages 49119

Elimination Theory
Pages 121174

The Algebra–Geometry Dictionary
Pages 175232

Polynomial and Rational Functions on a Variety
Pages 233289

Table of contents (11 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Ideals, Varieties, and Algorithms
 Book Subtitle
 An Introduction to Computational Algebraic Geometry and Commutative Algebra
 Authors

 David A Cox
 John Little
 Donal Oshea
 Series Title
 Undergraduate Texts in Mathematics
 Copyright
 2015
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer International Publishing Switzerland
 eBook ISBN
 9783319167213
 DOI
 10.1007/9783319167213
 Hardcover ISBN
 9783319167206
 Softcover ISBN
 9783319374277
 Series ISSN
 01726056
 Edition Number
 4
 Number of Pages
 XVI, 646
 Number of Illustrations
 88 b/w illustrations, 7 illustrations in colour
 Topics