Elementary Geometry of Algebraic Curves: An Undergraduate Introduction

ISBN-13: 9780521646413 / Angielski / Miękka / 1998 / 268 str.

C. G. Gibson (University of Liverpool)

Opis

Here is an introduction to plane algebraic curves from a geometric viewpoint, designed as a first text for undergraduates in mathematics, or for postgraduate and research workers in the engineering and physical sciences. The book is well illustrated and contains several hundred worked examples and exercises. From the familiar lines and conics of elementary geometry the reader proceeds to general curves in the real affine plane, with excursions to more general fields to illustrate applications, such as number theory. By adding points at infinity the affine plane is extended to the projective plane, yielding a natural setting for curves and providing a flood of illumination into the underlying geometry. A minimal amount of algebra leads to the famous theorem of Bezout, while the ideas of linear systems are used to discuss the classical group structure on the cubic.

Recenzje

"This book amply fulfills the promise of its title...far less forbidding than the vast majority of more ambitious textbooks...the author clearly motivates the study of projective curves by showing that several affine problems are more easily studied via the projective setting...a good abstract introduction for mathematicians." Theory of Computation

Spis treści

List of illustrations; List of tables; Preface; 1. Real algebraic curves; 2. General ground fields; 3. Polynomial algebra; 4. Affine equivalence; 5. Affine conics; 6. Singularities of affine curves; 7. Tangents to affine curves; 8. Rational affine curves; 9. Projective algebraic curves; 10. Singularities of projective curves; 11. Projective equivalence; 12. Projective tangents; 13. Flexes; 14. Intersections of projective curves; 15. Projective cubics; 16. Linear systems; 17. The group structure on a cubic; 18. Rational projective curves; Index.