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A theorem of Pólya on polynomials

  • Martin Aigner
  • Günter M. Ziegler

Abstract

Among the many contributions of Pólya to analysis, the following has always been Erdős’ favorite, both for the surprising result and for the beauty of its proof. Suppose that
$$f\left( z \right) = z^n + b_{n - 1} z^{n - 1} + \ldots + b_0 $$
is a complex polynomial of degree n ≥ 1 with leading coefficient 1. Associate with f(z) the set
$$C: = \{ z \in \mathbb{C}:|f\left( z \right)| \leq 2\},$$
that is, C is the set of points which are mapped under f 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.

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References

  1. [1]
    P. L. Cebycev: Œuvres, Vol. I, Acad. Imperiale des Sciences, St. Peter burg 1899, pp. 387-469.Google Scholar
  2. [2]
    G. Pőlya: Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfai zusammenhängenden Gebieten, Sitzungsber. Preuss. Akad. Wiss. Berl (1928), 228-232; Collected Papers Vol. I, MIT Press 1974, 347-351.Google Scholar
  3. [3]
    G. Pőlya & G. Szegő: Problems and Theorems in Analysis, Vol. II, Springe Verlag, Berlin Heidelberg New York 1976; Reprint 1998.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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