New estimates for the length of the Erdos-Herzog-Piranian lemniscate
Alexander Fryntov, Fedor Nazarov

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).
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Taxonomy
TopicsAnalytic Number Theory Research · Mathematical functions and polynomials · Advanced Combinatorial Mathematics
