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Undergraduate Texts in Mathematics

Ideals, Varieties, and Algorithms

An Introduction to Computational Algebraic Geometry and Commutative Algebra

Authors: Cox, David A, Little, John, O'Shea, Donal

  • 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
  • Presents some of the newer approaches to computing Groebner bases
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Buy this book

eBook $19.99
$54.99 (listprice)
price for USA (gross)
valid through June 30, 2017
  • ISBN 978-3-319-16721-3
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $29.99
$69.99 (listprice)
price for USA
valid through June 30, 2017
  • ISBN 978-3-319-16720-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $29.99
$69.99 (listprice)
price for USA
valid through June 30, 2017
  • Customers within the U.S. and Canada please contact Customer Service at 1-800-777-4643, Latin America please contact us at +1-212-460-1500 (Weekdays 8:30am – 5:30pm ET) to place your order.
  • Due: November 3, 2016
  • ISBN 978-3-319-37427-7
  • Free shipping for individuals worldwide
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 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.

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

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 (10 chapters)

  • Geometry, Algebra, and Algorithms

    Cox, David A. (et al.)

    Pages 1-47

  • Gröbner Bases

    Cox, David A. (et al.)

    Pages 49-119

  • Elimination Theory

    Cox, David A. (et al.)

    Pages 121-174

  • The Algebra–Geometry Dictionary

    Cox, David A. (et al.)

    Pages 175-232

  • Polynomial and Rational Functions on a Variety

    Cox, David A. (et al.)

    Pages 233-289

Buy this book

eBook $19.99
$54.99 (listprice)
price for USA (gross)
valid through June 30, 2017
  • ISBN 978-3-319-16721-3
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $29.99
$69.99 (listprice)
price for USA
valid through June 30, 2017
  • ISBN 978-3-319-16720-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $29.99
$69.99 (listprice)
price for USA
valid through June 30, 2017
  • Customers within the U.S. and Canada please contact Customer Service at 1-800-777-4643, Latin America please contact us at +1-212-460-1500 (Weekdays 8:30am – 5:30pm ET) to place your order.
  • Due: November 3, 2016
  • ISBN 978-3-319-37427-7
  • Free shipping for individuals worldwide
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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
Series Title
Undergraduate Texts in Mathematics
Copyright
2015
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-16721-3
DOI
10.1007/978-3-319-16721-3
Hardcover ISBN
978-3-319-16720-6
Softcover ISBN
978-3-319-37427-7
Series ISSN
0172-6056
Edition Number
4
Number of Pages
XVI, 646
Number of Illustrations and Tables
85 b/w illustrations, 10 illustrations in colour
Topics