Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics), by David A Cox, John Little, Donal O'Shea
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Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics), by David A Cox, John Little, Donal O'Shea
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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
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics), by David A Cox, John Little, Donal O'Shea- Amazon Sales Rank: #1438342 in eBooks
- Published on: 2015-06-14
- Released on: 2015-06-14
- Format: Kindle eBook
Review
From the reviews of the third edition:
"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 a excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry." (Peter Schenzel, Zentralblatt MATH, Vol. 1118 (20), 2007)
From the Back Cover
Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated?
The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.
The algorithms to answer questions such as those posed above are an important part of algebraic geometry. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.
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 includes:
A significantly updated section on Maple in Appendix C
Updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR
A shorter proof of the Extension Theorem presented in Section 6 of Chapter 3
From the 2nd Edition:
"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 Author
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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Most helpful customer reviews
1 of 1 people found the following review helpful. I guess the third edition is better. I don't like the new version of proof ... By S. Chen I guess the third edition is better. I don't like the new version of proof of extension theorem and Hilbert Nullstellensatz theorem. Proofs in the third edition are more elegant (at least to me). However, this book is by far the best to introduce us the subject with minimal prerequisites. It would be great if you take a course with this book as the textbook.
4 of 5 people found the following review helpful. Excellent introduction By Narada Certainly will take you from a novice to a high level of understanding. If you are already a professional mathematician, you will need to skim parts (the authors repeat things a lot, which is a very useful pedagogical device, but can be annoying if you are reading the book sequentially.
1 of 1 people found the following review helpful. Great follow-up to Advanced Alg By Ronald David Thurner Filled with great information about its subject (see title). Its my second copy. I ruined the first copy by working it so hard.
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