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Prove that there exist a constant $C$ such that for every monic polynomial $P$, the area of the set $A=\{x : |P(x)|<1\}$ is at most $C$.

Remarks:

  1. This puzzle holds for both the real and the complex field (with possibly different $C$'s)
  2. Monic polynomial means that the leading coefficient is 1
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  • $\begingroup$ Just look up Remez inequality for the real case and the Cartan lemma for the complex case. $\endgroup$ – fedja Feb 15 '13 at 14:08
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This is a famous result by Pólya.

G. Pólya, Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete, Sitzungsberichte Akad. Wiss. Berlin (1928), 228–232 & 280–282. Also in volume 1 of Pólya’s Collected Papers, MIT press, (1974).

For a modern account and a generalization, see Edward Crane, The Area of Polynomial Images and Preimages.

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