Number of components of polynomial lemniscates: a problem of Erdös, Herzog, and Piranian

Abstract

Let $K\subset \mathbb{C}$ be a compact set in the plane whose logarithmic capacity $c(K)$ is strictly positive. Let ${\mathcal{P}}_n(K)$ be the space of monic polynomials of degree $n,$ \emph{all} of whose zeros lie in $K.$ For $p\in {\mathcal{P}}_n(K),$ its filled \emph{unit leminscate} is defined by ${\Lambda }_p=\{z:|p(z)|<1\}.$ Let $\mathcal{C}({\Lambda }_p)$ denote the number of connected components of the open set ${\Lambda }_p,$ and define ${\mathcal{C}}_n(K)={\max }_{p\in {\mathcal{P}}_n(K)}\mathcal{C}({\Lambda }_p).$ In this paper we show that the quantity \[M(K) = \limsup_{n\to\infty}\dfrac{\mathscr{C}_n(K)}{n},\] satisfies $M(K)<1$ when the logarithmic capacity $c(K)<1,$ and $M(K)=1$ when $c(K)\ge 1.$ In particular, this answers a question of Erdös et. al. posed in $1958$. In addition, we show that for nice enough compact sets whose capacity is strictly bigger than $\frac{1}{2}$, the quantity $m(K)={\mathrm{liminf}}_{n\to \infty }\frac{{\mathcal{C}}_n(K)}{n}>0.$

20 pages, 3 figures