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)


Geometry, Algebra, and Algorithms
Pages 146

Groebner Bases
Pages 47111

Elimination Theory
Pages 112166

The AlgebraGeometry Dictionary
Pages 167211

Polynomial and Rational Functions on a Variety
Pages 212260

Table of contents (9 chapters)
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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

 David Cox
 John Little
 Donal Oshea
 Series Title
 Undergraduate Texts in Mathematics
 Copyright
 1997
 Publisher
 SpringerVerlag New York
 Copyright Holder
 Springer Science+Business Media New York
 eBook ISBN
 9781475726930
 DOI
 10.1007/9781475726930
 Series ISSN
 01726056
 Edition Number
 2
 Number of Pages
 XIII, 538
 Number of Illustrations
 44 b/w illustrations
 Topics
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