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© 2006

Conics and Cubics

A Concrete Introduction to Algebraic Curves

Textbook

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

Table of contents

  1. Front Matter
    Pages i-viii
  2. Pages 69-126
  3. Pages 127-244
  4. Pages 245-339
  5. Back Matter
    Pages 340-347

About this book

Introduction

Conics and Cubics is an accessible introduction to algebraic curves. Its focus on curves of degree at most three keeps results tangible and proofs transparent. Theorems follow naturally from high school algebra and two key ideas: homogenous coordinates and intersection multiplicities.

By classifying irreducible cubics over the real numbers and proving that their points form Abelian groups, the book gives readers easy access to the study of elliptic curves. It includes a simple proof of Bezout's Theorem on the number of intersections of two curves.

The book is a text for a one-semester course on algebraic curves for junior-senior mathematics majors. The only prerequisite is first-year calculus.

The new edition introduces the deeper study of curves through parametrization by power series. Two uses of parametrizations are presented: counting multiple intersections of curves and proving the duality of curves and their envelopes.

About the first edition:

"The book...belongs in the admirable tradition of laying the foundations of a difficult and potentially abstract subject by means of concrete and accessible examples."

- Peter Giblin, MathSciNet

Keywords

Area Grad algebra algebraic curve calculus geometry mathematics proof theorem

Authors and affiliations

  1. 1.Department of MathematicsThe University of Michigan-FlintFlint

Bibliographic information

  • Book Title Conics and Cubics
  • Book Subtitle A Concrete Introduction to Algebraic Curves
  • Authors Robert Bix
  • Series Title Undergraduate Texts in Mathematics
  • DOI https://doi.org/10.1007/0-387-39273-4
  • Copyright Information Springer Science+Business Media, LLC 2006
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-0-387-31802-8
  • Softcover ISBN 978-1-4419-2178-9
  • eBook ISBN 978-0-387-39273-8
  • Series ISSN 0172-6056
  • Edition Number 2
  • Number of Pages VIII, 347
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Algebraic Geometry
    Geometry
    Numerical Analysis
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking

Reviews

"...This book therefore belongs to the admirable tradition of laying the foundations of a difficult and potentially abstract subject by means of concrete and accessible examples. ... Two major strengths of the book are its historical perspective, in the form of informative introductions to the chapters which give the main developments in non-technical language, and its exercises, which are numerous and interesting." Peter Giblin for MathSciNet

From the reviews of the second edition:

"Algebraic geometry is a hard subject. ... But could it, or at least some of it, be presented, at the undergraduate level? This book attempts to do that. ... At the beginning of each of the four chapters, the author provides a synopsis of the historical development of the subject. And within each section many exercises are provided for further discussion and illumination. ... And the author manages to keep things concrete. So, the end result is a book which is accessible … ." (Donald L. Vestal, MathDL – online, October, 2006)