This book studies the geometric theory of polynomials and rational functions in the plane. Any theory in the plane should make full use of the complex numbers and thus the early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology and analysis. In fact, 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, bearing in mind that such problems indicate the current limitations of our knowledge and present challenges for the future. However, theories also lead to solutions of some problems and several such solutions are given including a comprehensive account of the geometric convolution theory. This is an ideal reference for graduate students and researchers working in this area.

"...a very welcome addition to the literature dealing with polynomials. ...will be invaluable to researchers and accessible to non-experts as well. I recommend it enthusiastically." Mathematical Reviews

"As both focused study and a joyride across the expanse of modern mathematics, this compilation of results by Sheil-Small would prject polynomial mathematics as a cohesive subject all its own.... Highly recommended." Choice