Conics and Cubics: A Concrete Introduction to Algebraic Curves / Edition 2

by Robert Bix

Overview

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

Product Details

ISBN-13:	9781441921789
Publisher:	Springer New York
Publication date:	11/25/2010
Series:	Undergraduate Texts in Mathematics Series
Edition description:	Softcover reprint of hardcover 2nd ed. 2006
Pages:	347
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

Preface     vii
Intersections of Curves     1
Introduction and History     1
Intersections at the Origin     4
Homogeneous Coordinates     17
Intersections in Homogeneous Coordinates     31
Lines and Tangents     50
Conics     69
Introduction and History     69
Conics and Intersections     73
Pascal's Theorem     92
Envelopes of Conics     110
Cubics     127
Introduction and History     127
Flexes and Singular Points     134
Addition on Cubics     153
Complex Numbers     175
Bezout's Theorem     195
Hessians     211
Determining Cubics     229
Parametrizing Curves     245
Introduction and History     245
Parametrizations at the Origin     249
Parametrizations of General Form     276
Envelopes of Curves     304
References     340
Index     343

Editorial 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)

From the Publisher