On the Arithmetic of Bicritical Rational Functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910003114278912 |
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| author | Goksel, Vefa Jones, Rafe |
| author_facet | Goksel, Vefa Jones, Rafe |
| contents | Bicritical rational functions -- those with precisely two critical points -- include the well-studied families of unicritical polynomials and quadratic rational functions. In this article we lay out general foundations for studying arithmetic dynamical properties of bicritical rational functions, and prove new Galois-theoretic results for a family with special properties. We study the field of definition of the critical points, and give a normal form up to Möbius conjugacy over this field. As a corollary, we show that after a finite extension of the ground field, the arboreal Galois representation attached to a bicritical rational function injects into an iterated wreath product of cyclic groups. We then examine the family of quadratic $ϕ\in \mathbb{Q}(x)$ with critical points $γ_1$ and $γ_2$ such that $ϕ(γ_1) = γ_2$. Adapting methods of Odoni-Stoll in the polynomial case to rational functions, we show that the arboreal representation is surjective for an infinite subfamily. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20122 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Arithmetic of Bicritical Rational Functions Goksel, Vefa Jones, Rafe Number Theory 37P05, 37P15, 11R32 Bicritical rational functions -- those with precisely two critical points -- include the well-studied families of unicritical polynomials and quadratic rational functions. In this article we lay out general foundations for studying arithmetic dynamical properties of bicritical rational functions, and prove new Galois-theoretic results for a family with special properties. We study the field of definition of the critical points, and give a normal form up to Möbius conjugacy over this field. As a corollary, we show that after a finite extension of the ground field, the arboreal Galois representation attached to a bicritical rational function injects into an iterated wreath product of cyclic groups. We then examine the family of quadratic $ϕ\in \mathbb{Q}(x)$ with critical points $γ_1$ and $γ_2$ such that $ϕ(γ_1) = γ_2$. Adapting methods of Odoni-Stoll in the polynomial case to rational functions, we show that the arboreal representation is surjective for an infinite subfamily. |
| title | On the Arithmetic of Bicritical Rational Functions |
| topic | Number Theory 37P05, 37P15, 11R32 |
| url | https://arxiv.org/abs/2601.20122 |