Dynamical Galois groups of trinomials and Odoni’s conjecture. (English) Zbl 07094881

Looper, Nicole

Bull. Lond. Math. Soc. 51, No. 2, 278-292 (2019).

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