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arXiv · 0808.0717

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

Alexander Fryntov·Fedor Nazarov
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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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Alexander Fryntov, Fedor Nazarov. 2008-08-05. New estimates for the length of the Erdos-Herzog-Piranian lemniscate. https://arxiv.org/abs/0808.0717

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