Overview
- 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
Part of the book series: Undergraduate Texts in Mathematics (UTM)
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Table of contents (11 chapters)
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Front Matter
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Back Matter
Keywords
- CoCoA algebraic geometry
- Groebner basis
- Hilbert basis theorem
- Macaulay2 algebraic geometry
- Maple algebraic geometry
- Mathematica algebraic geometry
- Nullstellensatz
- Sage algebraic geometry
- algebraic geometry textbook adoption
- algorithms algebraic geometry
- computational algebraic geometry
- invariant theory
- projective geometry
About this book
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 levelcourses 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 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 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
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)
Authors and Affiliations
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.
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 O’Shea
Series Title: Undergraduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-319-16721-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2015
Hardcover ISBN: 978-3-319-16720-6Published: 13 May 2015
Softcover ISBN: 978-3-319-37427-7Published: 17 October 2016
eBook ISBN: 978-3-319-16721-3Published: 30 April 2015
Series ISSN: 0172-6056
Series E-ISSN: 2197-5604
Edition Number: 4
Number of Pages: XVI, 646
Number of Illustrations: 88 b/w illustrations, 7 illustrations in colour
Topics: Algebraic Geometry, Commutative Rings and Algebras, Mathematical Logic and Foundations, Mathematical Software