On the FOD/FOM parameter of rational maps
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915016131739648 |
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| author | Hidalgo, Ruben A. |
| author_facet | Hidalgo, Ruben A. |
| contents | Let $χ$ be a (right) action of ${\rm PSL}_{2}({\mathbb L})$ on the space ${\mathbb L}(z)$ of rational maps defined over an algebraically closed field ${\mathbb L}$. If $R \in {\mathbb L}(z)$ and ${\mathcal M}_{R}^χ$ is its $χ$-field of moduli, then the parameter ${\rm FOD/FOM}_χ(R)$ is the smallest integer $n \geq 1$ such that there is a $χ$-field of definition of $R$ being a degree $n$ extension of ${\mathcal M}_{R}^χ$. When ${\mathbb L}$ has characteristic zero and $χ=χ_{\infty}$ is the conjugation action, then it is known that ${\rm FOD/FOM}_{χ_{\infty}}(R) \leq 2$. In this paper, we study the above parameter for general actions and any characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08006 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the FOD/FOM parameter of rational maps Hidalgo, Ruben A. Dynamical Systems 37F10, 37P05, 30F30 Let $χ$ be a (right) action of ${\rm PSL}_{2}({\mathbb L})$ on the space ${\mathbb L}(z)$ of rational maps defined over an algebraically closed field ${\mathbb L}$. If $R \in {\mathbb L}(z)$ and ${\mathcal M}_{R}^χ$ is its $χ$-field of moduli, then the parameter ${\rm FOD/FOM}_χ(R)$ is the smallest integer $n \geq 1$ such that there is a $χ$-field of definition of $R$ being a degree $n$ extension of ${\mathcal M}_{R}^χ$. When ${\mathbb L}$ has characteristic zero and $χ=χ_{\infty}$ is the conjugation action, then it is known that ${\rm FOD/FOM}_{χ_{\infty}}(R) \leq 2$. In this paper, we study the above parameter for general actions and any characteristic. |
| title | On the FOD/FOM parameter of rational maps |
| topic | Dynamical Systems 37F10, 37P05, 30F30 |
| url | https://arxiv.org/abs/2411.08006 |