# Ideals, Varieties, and Algorithms

## An Introduction to Computational Algebraic Geometry and Commutative Algebra

• David Cox
• John Little
• Donal O’Shea
Book

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xiii
2. David Cox, John Little, Donal O’Shea
Pages 1-46
3. David Cox, John Little, Donal O’Shea
Pages 47-111
4. David Cox, John Little, Donal O’Shea
Pages 112-166
5. David Cox, John Little, Donal O’Shea
Pages 167-211
6. David Cox, John Little, Donal O’Shea
Pages 212-260
7. David Cox, John Little, Donal O’Shea
Pages 261-310
8. David Cox, John Little, Donal O’Shea
Pages 311-348
9. David Cox, John Little, Donal O’Shea
Pages 349-428
10. David Cox, John Little, Donal O’Shea
Pages 429-495
11. Back Matter
Pages 497-538

### Introduction

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.

### Keywords

#### Authors and affiliations

• David Cox
• 1
• John Little
• 2
• Donal O’Shea
• 3
1. 1.Department of Mathematics and Computer ScienceAmherst CollegeAmherstUSA
2. 2.Department of MathematicsCollege of the Holy CrossWorcesterUSA
3. 3.Department of Mathematics, Statistics, and Computer ScienceMount Holyoke CollegeSouth HadleyUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-2693-0
• Copyright Information Springer-Verlag New York 1997
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4757-2695-4
• Online ISBN 978-1-4757-2693-0
• Series Print ISSN 0172-6056