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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.

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