On the length of lemniscates

TL;DR

This paper establishes an upper bound on the length of lemniscates for monic polynomials, improving previous estimates, and characterizes the extremal case for quadratic polynomials as Bernoulli's lemniscate.

Contribution

It provides a new upper bound for the length of lemniscates of monic polynomials and characterizes the extremal case for degree two.

Findings

Length of lemniscates is at most 9.2 times the degree.

Extremal quadratic lemniscate is Bernoulli's lemniscate.

Extremal lemniscates are connected for polynomials achieving the maximum length.

Abstract

We show that for a monic polynomial p of degree d, the length of the level set {z: |p(z)|=1} is at most 9.2 d, which improves an earlier estimate due to P. Borwein. For d=2 we show that the extremal level set is the Bernoullis' Lemniscate. One ingredient of our proofs is the fact that for an extremal polynomial this level set is connected.