Ideals, Varieties, and Algorithms
An Introduction to Computational Algebraic Geometry and Commutative Algebra
Authors: Cox, David A, Little, John, Oshea, Donal
Free Preview Covers important topics such as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory
 Third edition includes a significantly updated section on Maple
 Contains updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica, and SINGULAR
 Presents a shorter proof of the Extension Theorem
 Over 200 pages have been revised in this third edition to enhance clarity and correctness
 Written at a level appropriate for undergraduates
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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. 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 2^{nd} 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
 Reviews

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 wellwritten. … 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)
 Table of contents (9 chapters)


Geometry, Algebra, and Algorithms
Pages 148

Groebner Bases
Pages 49114

Elimination Theory
Pages 115168

The Algebra–Geometry Dictionary
Pages 169214

Polynomial and Rational Functions on a Variety
Pages 215264

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 A Cox
 John Little
 Donal Oshea
 Series Title
 Undergraduate Texts in Mathematics
 Copyright
 2007
 Publisher
 SpringerVerlag New York
 Copyright Holder
 SpringerVerlag New York
 eBook ISBN
 9780387356518
 DOI
 10.1007/9780387356518
 Series ISSN
 01726056
 Edition Number
 3
 Number of Pages
 XV, 553
 Topics