On dessins d'enfants with equal supports
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913845459550208 |
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| author | Pakovich, Fedor |
| author_facet | Pakovich, Fedor |
| contents | For a Belyi function $β:\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ ramified only over the points $-1,1,\infty$, a corresponding ``dessin d'enfant'' $\mathcal D_β$ is defined as the set $β^{-1}([-1,1])$ considered as a bi-colored graph on the Riemann sphere whose white and black vertices are points of the sets $β^{-1}\{-1\}$ and $β^{-1}\{1\}$ correspondingly. Merely the set $β^{-1}([-1,1])$ without a graph structure is called a support of $\mathcal D_β$. In this note, we solve the following problem: under what conditions different dessins $\mathcal D_{β_1}$ and $\mathcal D_{β_2}$ have equal supports? |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On dessins d'enfants with equal supports Pakovich, Fedor Complex Variables Algebraic Geometry Number Theory For a Belyi function $β:\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ ramified only over the points $-1,1,\infty$, a corresponding ``dessin d'enfant'' $\mathcal D_β$ is defined as the set $β^{-1}([-1,1])$ considered as a bi-colored graph on the Riemann sphere whose white and black vertices are points of the sets $β^{-1}\{-1\}$ and $β^{-1}\{1\}$ correspondingly. Merely the set $β^{-1}([-1,1])$ without a graph structure is called a support of $\mathcal D_β$. In this note, we solve the following problem: under what conditions different dessins $\mathcal D_{β_1}$ and $\mathcal D_{β_2}$ have equal supports? |
| title | On dessins d'enfants with equal supports |
| topic | Complex Variables Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2505.12420 |