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

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## Description

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 Groebner 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 (TM), Mathematica (R) 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

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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 (TM), Mathematica (R) 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

show more

## Product details

- Hardback | 646 pages
- 155 x 235 x 36.58mm | 1,097.69g
- 22 May 2015
- Springer International Publishing AG
- Cham, Switzerland
- English
- Revised
- 4th ed. 2015
- 7 Illustrations, color; 88 Illustrations, black and white; XVI, 646 p. 95 illus.
- 3319167200
- 9783319167206
- 595,790

## Back cover copy

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(TM), Mathematica(R), 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

show more

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(TM), Mathematica(R), 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

show more

## Table of contents

Preface.- Notation for Sets and Functions.- 1. Geometry, Algebra, and Algorithms.- 2. Groebner Bases.- 3. Elimination Theory.- 4.The Algebra-Geometry Dictionary.- 5. Polynomial and Rational Functions on a Variety.- 6. Robotics and Automatic Geometric Theorem Proving.- 7. Invariant Theory of Finite Groups.- 8. Projective Algebraic Geometry.- 9. The Dimension of a Variety.- 10. Additional Groebner Basis Algorithms.- Appendix A. Some Concepts from Algebra.- Appendix B. Pseudocode.- Appendix C. Computer Algebra Systems.- Appendix D. Independent Projects.- References.- Index.

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## Review Text

"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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## Review quote

"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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## About David A. Cox

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.

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