Many lemniscates with large diameter
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908540382216192 |
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| author | Huang, Linhang |
| author_facet | Huang, Linhang |
| contents | We prove that for every $0 < c < 4$ and every $N \in \mathbb{N}$ there exists a monic polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0$ such that the set $\{z \in \mathbb{C} : |p(z)| \leq 1\}$ has at least $N$ connected components with diameter at least $c$. This answers a question of Erdős. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Many lemniscates with large diameter Huang, Linhang Complex Variables We prove that for every $0 < c < 4$ and every $N \in \mathbb{N}$ there exists a monic polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0$ such that the set $\{z \in \mathbb{C} : |p(z)| \leq 1\}$ has at least $N$ connected components with diameter at least $c$. This answers a question of Erdős. |
| title | Many lemniscates with large diameter |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2509.11597 |