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

Abstract

Let K⊂C be a compact set in the plane whose logarithmic capacity c(K) is strictly positive. Let Pn(K) be the space of monic polynomials of degree n, all of whose zeros lie in K. For p∈Pn(K), its filled unit leminscate is defined by Λp={z:|p(z)|<1}. Let C(Λp) denote the number of connected components of the open set Λp, and define Cn(K)=maxp∈Pn(K)⁡C(Λp). In this paper we show that the quantityM(K)=limsupn→∞Cn(K)n, satisfies M(K)<1 when the logarithmic capacity c(K)<1, and M(K)=1 when c(K)≥1. In particular, this answers a question of Erdös et al. posed in (1958) [8]. In addition, we show that for nice enough compact sets whose capacity is strictly bigger than 12, the quantity m(K)=liminfn→∞Cn(K)n>0.