A Guide to Plane Algebraic Curves: (Dolciani Mathematical Expositions)

This is an informal and accessible introduction to plane algebraic curves that also serves as a natural entry point to algebraic geometry. There is a unifying theme to the book: give curves enough living space and beautiful theorems will follow. This book provides the reader with a solid intuition for the subject, while at the same time keeping the exposition simple and understandable, by introducing abstract concepts with concrete examples and pictures. It can be used as the text for an undergraduate course on plane algebraic curves, or as a companion to algebraic geometry at graduate level. This book is accessible to those with a limited mathematical background. This is because for those outside mathematics there is a growing need for an entrée to algebraic geometry, a need created by the ever-expanding role algebraic geometry is playing in areas ranging from biology to chemistry and robotics to cryptology.

Table of Contents:
Preface; 1. A gallery of algebraic curves; 2. Points at infinity; 3. From real to complex; 4. Topology of algebraic curves in P2.C/; 5. Singularities; 6. The big three: C, K, S; Bibliography.

About the Author :
Keith Kendig is a Professor of Mathematics at Cleveland State University.

Review :
Algebraic curves have regained a prominent position in mathematics. In light of their importance, the goal of this book is to provide a reasonable understanding of algebraic curves and their use. Beginning with standard curves (polynomial, parametric, conic, and user defined), Kendig expands the study by first shrinking the plane to a disk by adjoining points at infinity, and then shifting the domain from real to complex numbers to establish Bezout's theorem. Given this context, the study shifts further to determining the topological properties of algebraic curves, relating genus to a polynomial's degree, investigating singularities, and using compact Riemann surfaces. Throughout, the author emphasizes the geometry and intuitive aspects of algebraic curves, without delving into a tedious chain of proofs. He briefly considers applications of algebraic curves, ranging from Andrew Wiles's special use of elliptic curves in his proof of Fermat's last theorem to their use in cryptography, dynamical systems, and robotics. Readers should be familiar with basic ideas from geometric topology, complex analysis, and abstract algebra. Since Kendig developed the content as a guide rather than a textbook, no problem sets are included, but the author does suggest appropriate textbooks in a bibliography."" - J. Johnson, CHOICE Magazine ""This book is a straightforward and simple introduction to plane algebraic curves and would be of interest to anyone wanting a good overview of the subject. The material could also serve as a useful supplement for students taking an introductory course on algebraic curves or for mathematicians who would like to learn about the subject. Even though the book is written as a guide, readers will need some basic understanding of complex analysis, field theory, and topology to comprehend the subject matter completely. The book is informal in its approach and focuses on building intuition through a variety of pictures and clarifying examples without becoming mired in detailed proofs and exercises. Chapter 1 explores algebraic curves in the real plane; in subsequent chapters, the canvas expands to include the complex numbers. The book's final chapters focus more on the geometric properties of algebraic curves and conclude with a foray into the topic of Riemann surfaces. A Guide to Plane Algebraic Curves is an accessible and well-written book that anyone with an interest in this beautiful subject will surely appreciate and find useful."" - Marc Michael, Mathematics Teacher