Arboreal Galois groups for quadratic rational functions with colliding critical points
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914819919052800 |
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| author | Benedetto, Robert L. Dietrich, Anna |
| author_facet | Benedetto, Robert L. Dietrich, Anna |
| contents | Let $K$ be a field, and let $f\in K(z)$ be rational function. The preimages of a point $x_0\in P^1(K)$ under iterates of $f$ have a natural tree structure. As a result, the Galois group of the resulting field extension of $K$ naturally embeds into the automorphism group of this tree. In unpublished work from 2013, Pink described a certain proper subgroup $M_{\ell}$ that this so-called arboreal Galois group $G_{\infty}$ must lie in if $f$ is quadratic and its two critical points collide at the $\ell$-th iteration. After presenting a new description of $M_{\ell}$ and a new proof of Pink's theorem, we state and prove necessary and sufficient conditions for $G_{\infty}$ to be the full group $M_{\ell}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16284 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Arboreal Galois groups for quadratic rational functions with colliding critical points Benedetto, Robert L. Dietrich, Anna Number Theory Dynamical Systems Primary: 37P05. Secondary: 11R32, 14G25 Let $K$ be a field, and let $f\in K(z)$ be rational function. The preimages of a point $x_0\in P^1(K)$ under iterates of $f$ have a natural tree structure. As a result, the Galois group of the resulting field extension of $K$ naturally embeds into the automorphism group of this tree. In unpublished work from 2013, Pink described a certain proper subgroup $M_{\ell}$ that this so-called arboreal Galois group $G_{\infty}$ must lie in if $f$ is quadratic and its two critical points collide at the $\ell$-th iteration. After presenting a new description of $M_{\ell}$ and a new proof of Pink's theorem, we state and prove necessary and sufficient conditions for $G_{\infty}$ to be the full group $M_{\ell}$. |
| title | Arboreal Galois groups for quadratic rational functions with colliding critical points |
| topic | Number Theory Dynamical Systems Primary: 37P05. Secondary: 11R32, 14G25 |
| url | https://arxiv.org/abs/2307.16284 |