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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2512.12455 |
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| _version_ | 1866912781676052480 |
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| author | Tao, Terence |
| author_facet | Tao, Terence |
| contents | Let $n \geq 1$, and let $p : {\bf C} \to {\bf C}$ be a monic polynomial of degree $n$. It was conjectured by Erdős, 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The maximal length of the Erdős--Herzog--Piranian lemniscate in high degree Tao, Terence Complex Variables 30C75 Let $n \geq 1$, and let $p : {\bf C} \to {\bf C}$ be a monic polynomial of degree $n$. It was conjectured by Erdős, 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 $n$. |
| title | The maximal length of the Erdős--Herzog--Piranian lemniscate in high degree |
| topic | Complex Variables 30C75 |
| url | https://arxiv.org/abs/2512.12455 |