Polynomials play an important role in almost all areas of mathematics. From finding solutions to equations, finding the number of solutions of equations, understanding the role of critical points in determining the geometric behaviour of the distribution of values, determining the properties of geometric curves and much else, polynomials have wielded an enormous influence on the development of mathematics since ancient times. This is a book on polynomials in the plane with a special emphasis on the geometric theory. We can do justice to only a very small part of the subject and therefore we confine most of our attention to the study of a number of specific problems (some solved and some unsolved). However, as the book is directed towards graduate students and to a broad audience of scientists and mathematicians possessing a basic knowledge of complex variable theory, we have concentrated in the earlier chapters on building some theoretical foundations, melding together algebraic, topological and analytic ideas.

Description of the chapters.

Chapter 1. The algebra of polynomials. The chapter deals with two important issues concerning real analytic polynomials – a topic normally regarded as being part of classical algebraic geometry. The first is a straightforward account of Bézout's theorem in the plane. We prove a version of the theorem which is independent of any algebraic restrictions, and therefore immediately applicable and useful to the analyst. The second topic concerns the calculation and properties of asymptotic values at infinity.