The maximal length of the Erdős--Herzog--Piranian lemniscate in high degree
Authors
Terence Tao
Abstract
Let $n \geq 1$, and let $p : {\bf C} \to {\bf C}$ be a monic polynomial of degree $n$. It was conjectured by Erdős, Herzog, and Piranian that the maximal length of lemniscate $\{z \in {\bf C}: |p(z)| = 1\}$ is attained by the polynomial $p(z) = z^n-1$. In this paper, building upon a previous analysis of Fryntov and Nazarov, we establish this conjecture for all sufficiently large $n$.
