Abstract
Let P(z) be a monic polynomial of degree n, and α, ε>0. A classic lemma of Cartan asserts that the lemniscate E(P;ε):={z:|P(z)|≤εn} can be covered by balls Bj,1≤j≤n, whose diameters d(Bj) satisfy
∑j=1p(d(Bj))α≤e(4ε)α.
For a=2, this shows that E(P;ε) has an area at most πe(2ε)2. Pólya showed in this case that the sharp estimate is πε2. We discuss some of the ramifications of these estimates, as well as some of their close cousins, for example when P is normalized to have Lp norm 1 on some circle, and Remez’ inequality.