On the area of polynomial lemniscates

Abstract: Erd\"os posed in 1940 the extremal problem of studying the minimal area of the lemniscate ${|p(z)|<1}$ of a monic polynomial $p$ of degree $n$ all of whose zeros are in the closed unit disc. In this article, we prove that there exist positive constants $c,C$ independent of the degree $n$ such that [ \dfrac{c}{\log n} \leq \min \text{Area}( { |p(z)|<1 } ) \leq \frac{C}{\log \log n},] improving substantially the previously best known lower bound (due to Pommerenke in 1961) as well as improving the best known upper bound (due to Wagner in 1988). We also study the inradius (radius of the largest inscribed disc); we provide an estimate for the inradius in terms of the area that confirms a 2009 conjecture of Solynin and Williams, and we use this to give a lower bound of order $(n \sqrt{\log n})^{-1}$ on the inradius, addressing a 1958 problem posed by Erd\"os, Herzog, and Piranian (confirming their conjecture up to the logarithmic factor). In addition to studying the area of ${|p(z)|<1}$, we consider other sublevel sets ${|p(z)|<t\}$, proving both upper and lower bounds of the same order $1/\log \log n$ when $t\>1$ and proving power law upper and lower bounds when $0<t<1$. We also consider the minimal area problem under a more general constraint, namely, replacing the unit disc with a compact set $K$ of unit capacity, where we show that the minimal area converges to zero as $n \rightarrow \infty$ (giving an affirmative answer to another question of Erd\"os, Herzog, Piranian); we also investigate the structure of the area minimizing polynomials, showing that the normalized zero-counting measure converges to the equilibrium measure of $K$ as the degree $n \rightarrow \infty$.