# The area of the set in which a polynomial is “small”

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
• Just look up Remez inequality for the real case and the Cartan lemma for the complex case. – fedja Feb 15 '13 at 14:08