2004 | OriginalPaper | Chapter
A theorem of Pólya on polynomials
Authors : Martin Aigner, Günter M. Ziegler
Published in: Proofs from THE BOOK
Publisher: Springer Berlin Heidelberg
Included in: Professional Book Archive
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Among the many contributions of George Pólya to analysis, the following has always been Erdős’ favorite, both for the surprising result and for the beauty of its proof. Suppose that $$ f(z) = {z^n} + {b_{n - 1}}{z^{n - 1}} + \cdots + {b_0}$$ is a complex polynomial of degree n ≥ 1 with leading coefficient 1. Associate with f(z) the set $$ C: = \left\{ {z \in :\left| {f(z)} \right| \le 2} \right\}$$ that is, Cis the set of points which are mapped under / into the circle of radius 2 around the origin in the complex plane. So for n = 1 the domain C is just a circular disk of diameter 4.