Ideals, Varieties, and Algorithms

An Introduction to Computational Algebraic Geometry and Commutative Algebra

  • David Cox
  • John Little
  • Donal O’Shea

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

Table of contents

  1. Front Matter
    Pages i-xi
  2. David Cox, John Little, Donal O’Shea
    Pages 1-47
  3. David Cox, John Little, Donal O’Shea
    Pages 48-112
  4. David Cox, John Little, Donal O’Shea
    Pages 113-167
  5. David Cox, John Little, Donal O’Shea
    Pages 168-212
  6. David Cox, John Little, Donal O’Shea
    Pages 213-254
  7. David Cox, John Little, Donal O’Shea
    Pages 255-305
  8. David Cox, John Little, Donal O’Shea
    Pages 306-344
  9. David Cox, John Little, Donal O’Shea
    Pages 345-408
  10. David Cox, John Little, Donal O’Shea
    Pages 409-477
  11. Back Matter
    Pages 479-514

About this book

Introduction

We wrote this book to introduce undergraduates to some interesting ideas in algebraic geometry and commutative algebra. Until recently, these topics involved a lot of abstract mathematics and were only taught in graduate school. But in the 1960's, Buchberger and Hironaka discovered new algorithms for manipulating systems of polynomial equations. Fueled by the development of computers fast enough to run these algorithms, the last two decades have seen a minor revolution in commutative algebra. The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand, and has changed the practice of much research in algebraic geometry. This has also enhanced the importance of the subject for computer scientists and engineers, who have begun to use these techniques in a whole range of problems. It is our belief that the growing importance of these computational techniques warrants their introduction into the undergraduate (and graduate) mathematics curricu­ lum. Many undergraduates enjoy the concrete, almost nineteenth century, flavor that a computational emphasis brings to the subject. At the same time, one can do some substantial mathematics, including the Hilbert Basis Theorem, Elimination Theory and the Nullstellensatz. The mathematical prerequisites of the book are modest: the students should have had a course in linear algebra and a course where they learned how to do proofs. Examples of the latter sort of course include discrete math and abstract algebra.

Keywords

Dimension Division Grad algebraic geometry algorithms boundary element method commutative property computation geometry proving robot robotics theorem theorem proving variable

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/10.1007/978-1-4757-2181-2
  • Copyright Information Springer-Verlag New York 1992
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4757-2183-6
  • Online ISBN 978-1-4757-2181-2
  • Series Print ISSN 0172-6056
  • About this book