New estimates for the length of the Erdos-Herzog-Piranian lemniscate

TL;DR

This paper investigates the length of the lemniscate defined by monic polynomials, establishing a local maximum at the polynomial z^n-1 and providing an asymptotically sharp upper bound.

Contribution

It proves that the lemniscate length reaches a local maximum at z^n-1 and derives an asymptotically sharp bound of 2n+o(n) for its length.

Findings

Length of lemniscate has a local maximum at p(z)=z^n-1.

Established an asymptotically sharp upper bound of 2n+o(n).

Supports the conjecture about maximal lemniscate length at z^n-1.

Abstract

Let p(z) be a monic polynomial of degree n. Consider the lemniscate L={z:|p(z)|=1}. It has been conjectured that L has the largest length when p(z)=z^n-1. We show that the length of L attains a local maximum at this polynomial and prove the asymptotically sharp bound |L|<2n+o(n).