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arXiv · 2512.12455

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

Terence Tao
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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 $n$.

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Terence Tao. 2025-12-13. The maximal length of the Erd\H{o}s--Herzog--Piranian lemniscate in high degree. https://arxiv.org/abs/2512.12455

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