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Undergraduate Texts in Mathematics

Ideals, Varieties, and Algorithms

An Introduction to Computational Algebraic Geometry and Commutative Algebra

Authors: Cox, David, Little, John, Oshea, Donal

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About this Textbook

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. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing a new edition of Ideals, Varieties and Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and REDUCE.

Reviews

"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

Table of contents (9 chapters)

Table of contents (9 chapters)

Buy this book

eBook $74.99
price for USA in USD
  • ISBN 978-1-4757-2693-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
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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
1997
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4757-2693-0
DOI
10.1007/978-1-4757-2693-0
Series ISSN
0172-6056
Edition Number
2
Number of Pages
XIII, 538
Number of Illustrations
44 b/w illustrations
Topics

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