Dynamical Galois groups of trinomials and Odoni's conjecture

TL;DR

This paper proves Odoni's conjecture for all prime degrees by constructing specific polynomials with surjective arboreal Galois representations, and discusses implications of Vojta's conjecture for non-prime degrees.

Contribution

It establishes Odoni's conjecture for prime degrees and links Vojta's conjecture to the existence of such polynomials in non-prime degrees.

Findings

Odoni's conjecture proven for all prime degrees

Existence of polynomials with surjective arboreal Galois representations in prime degrees

Vojta's conjecture implies similar existence in many non-prime degrees

Abstract

We prove Odoni's conjecture in all prime degrees; namely, we prove that for every positive prime $p$, there exists a degree $p$ polynomial $\varphi\in\mathbb{Z}[x]$ with surjective arboreal Galois representation. We also show that Vojta's conjecture implies the existence of such a polynomial in many degrees $d$ which are not prime.