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Large arboreal Galois representations. (English) Zbl 07161186
Summary: Given a field \(K\), a polynomial \(f \in K [x]\) of degree \(d\), and a suitable element \(t \in K\), the set of preimages of \(t\) under the iterates \(f^{\circ n}\) carries a natural structure of a \(d\)-ary tree. We study conditions under which the absolute Galois group of \(K\) acts on the tree by the full group of automorphisms. When \(d \geq 20\) is even and \(K = \mathbb{Q}\) we exhibit examples of polynomials with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of \(K = F(t)\) and \(f \in F [x]\) in which the corresponding Galois groups are the monodromy groups of the ramified covers \(f^{\circ n} : \mathbb{P}_F^1 \rightarrow \mathbb{P}_F^1\).

MSC:
11R32 Galois theory
11F80 Galois representations
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
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