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

✍ Scribed by David A. Cox, John Little, Donal O’Shea

• Publisher: Springer-Verlag New York
• Year: 2007
• Tongue: English
• Leaves: 593
• Series: Undergraduate Texts in Mathematics
• Edition: 3
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Front Matter....Pages i-xv
Geometry, Algebra, and Algorithms....Pages 1-48
Groebner Bases....Pages 49-114
Elimination Theory....Pages 115-168
The Algebra–Geometry Dictionary....Pages 169-214
Polynomial and Rational Functions on a Variety....Pages 215-264
Robotics and Automatic Geometric Theorem Proving....Pages 265-316
Invariant Theory of Finite Groups....Pages 317-356
Projective Algebraic Geometry....Pages 357-438
The Dimension of a Variety....Pages 439-508
Back Matter....Pages 509-551

✦ Subjects

Algebraic Geometry; Commutative Rings and Algebras; Mathematical Logic and Foundations; Mathematical Software