The maximal length of the Erd\H{o}s--Herzog--Piranian lemniscate in high degree

TL;DR

This paper proves that for large degrees, the polynomial p(z) = z^n - 1 maximizes the length of its lemniscate among all monic polynomials of degree n, confirming a longstanding conjecture.

Contribution

It confirms the Erd ext{"o}s--Herzog--Piranian conjecture for all sufficiently large degrees, extending previous partial results.

Findings

The length of the lemniscate is maximized by p(z) = z^n - 1 for large n.

The conjecture holds asymptotically as n approaches infinity.

The analysis builds upon and extends Fryntov and Nazarov's previous work.

Abstract

Let $n \geq 1$, and let $p : {\bf C} \to {\bf C}$ be a monic polynomial of degree $n$. It was conjectured by Erd\H{o}s, 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 $n$.