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

✍ Scribed by Cox, David A;Little, John B;O'Shea, Donal

Publisher: Springer
Year: 2007
Tongue: English
Leaves: 564
Series: Undergraduate Texts in Mathematics
Category: Library

No coin nor oath required. For personal study only.

✦ 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 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 MonthlyP>

✦ Table of Contents

Preface to the First Edition......Page 7
Preface to the Second Edition......Page 9
Preface to the Third Edition......Page 11
Contents......Page 12
1 Geometry, Algebra, and Algorithms......Page 15
2 Groebner Bases......Page 63
3 Elimination Theory......Page 129
4 The Algebra-Geometry Dictionary......Page 183
5 Polynomial and Rational Functions on a Variety......Page 229
6 Robotics and Automatic Geometric Theorem Proving......Page 279
7 Invariant Theory of Finite Groups......Page 331
8 Projective Algebraic Geometry......Page 371
9 The Dimension of a Variety......Page 453
A Some Concepts from Algebra......Page 523
B Pseudocode......Page 527
C Computer Algebra Systems......Page 531
D Independent Projects......Page 544
References......Page 549
Index......Page 554