Unicritical polynomials over $abc$-fields: from uniform boundedness to dynamical Galois groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909378807857152 |
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| author | Doyle, John R. Hindes, Wade |
| author_facet | Doyle, John R. Hindes, Wade |
| contents | Let $K$ be a function field of characteristic $p\geq0$ or a number field over which the $abc$ conjecture holds, and let $ϕ(x)=x^d+c \in K[x]$ be a unicritical polynomial of degree $d\geq2$ with $d \not\equiv 0,1\pmod{p}$. We completely classify all portraits of $K$-rational preperiodic points for such $ϕ$ for all sufficiently large degrees $d$. More precisely, we prove that, up to accounting for the natural action of $d$th roots of unity on the preperiodic points for $ϕ$, there are exactly thirteen such portraits up to isomorphism. In particular, for all such global fields $K$, it follows from our results together with earlier work of Doyle-Poonen and Looper that the number of $K$-rational preperiodic points for $ϕ$ is uniformly bounded -- independent of $d$. That is, there is a constant $B(K)$ depending only on $K$ such that \[\big|\text{PrePer}(x^d+c,K)\big|\leq B(K)\] for all $d\geq2$ and all $c\in K$. Moreover, we apply this work to construct many irreducible polynomials with large dynamical Galois groups in semigroups generated by sets of unicritical polynomials under composition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14657 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unicritical polynomials over $abc$-fields: from uniform boundedness to dynamical Galois groups Doyle, John R. Hindes, Wade Number Theory Dynamical Systems Let $K$ be a function field of characteristic $p\geq0$ or a number field over which the $abc$ conjecture holds, and let $ϕ(x)=x^d+c \in K[x]$ be a unicritical polynomial of degree $d\geq2$ with $d \not\equiv 0,1\pmod{p}$. We completely classify all portraits of $K$-rational preperiodic points for such $ϕ$ for all sufficiently large degrees $d$. More precisely, we prove that, up to accounting for the natural action of $d$th roots of unity on the preperiodic points for $ϕ$, there are exactly thirteen such portraits up to isomorphism. In particular, for all such global fields $K$, it follows from our results together with earlier work of Doyle-Poonen and Looper that the number of $K$-rational preperiodic points for $ϕ$ is uniformly bounded -- independent of $d$. That is, there is a constant $B(K)$ depending only on $K$ such that \[\big|\text{PrePer}(x^d+c,K)\big|\leq B(K)\] for all $d\geq2$ and all $c\in K$. Moreover, we apply this work to construct many irreducible polynomials with large dynamical Galois groups in semigroups generated by sets of unicritical polynomials under composition. |
| title | Unicritical polynomials over $abc$-fields: from uniform boundedness to dynamical Galois groups |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2408.14657 |