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Dynamical Galois groups of trinomials and Odoni’s conjecture. (English) Zbl 07094881
Summary: We prove that for every prime \(p\), there exists a degree \(p\) polynomial whose arboreal Galois representation is surjective, that is, whose iterates have Galois groups over \(\mathbb{Q}\) that are as large as possible subject to a natural constraint coming from iteration. This resolves in the case of prime degree a conjecture of Odoni from 1985. We also show that a standard height uniformity conjecture in arithmetic geometry implies the existence of such a polynomial in many degrees \(d\) which are not prime.

11R32 Galois theory
37P15 Dynamical systems over global ground fields
14G05 Rational points
11D45 Counting solutions of Diophantine equations
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