Complex Polynomials [Hardcover]

This book studies the geometric theory of polynomials and rational functions in the plane.This book studies the geometric theory of polynomials and rational functions in the plane. The early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology and analysis. The theory is carefully contructed bearing in mind the needs of graduate students. Several unsolved problems are presented as well as the full solutions to some well known conjectures. As such this is an ideal reference for graduate students and researchers working in this area.This book studies the geometric theory of polynomials and rational functions in the plane. The early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology and analysis. The theory is carefully contructed bearing in mind the needs of graduate students. Several unsolved problems are presented as well as the full solutions to some well known conjectures. As such this is an ideal reference for graduate students and researchers working in this area.Complex Polynomials explores the geometric theory of polynomials and rational functions in the plane. Early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology, and analysis. Throughout the book, the author introduces a variety of ideas and constructs theories around them, incorporating much of the classical theory of polynomials as he proceeds. These ideas are used to study a number of unsolved problems. Several solutions to problems are given, including a comprehensive account of the geometric convolution theory.Preface; List of notation; 1. The algebra of polynomials; 2. The degree principle and the fundamental theorem of algebra; 3. The Jacobian problem; 4. Analytic and harmonic functions in the unit disc; 5. Circular regions and Grace's theorem; 6. The Ilieff-Sendov conjecture; 7. Self-inversive polynomials; 8. Duality and an extension of Gralӻ