Iterates of generic polynomials and generic rational functions

Abstract

Abstract. In 1985, Odoni showed that in characteristic 0 the Galois group of the n-th iterate of the generic polynomial with degree d is as large as possible. That is, he showed that this Galois group is the n-th wreath power of the symmetric group S d . We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.Several of the results proven in this paper were stated and proven by R.W.K. Odoni in an unpublished preprint, including the polynomial versions of the Galois theoretic results and the application presented in Section 6.2. Although the results and arguments given by Odoni in that manuscript are presented somewhat differently here, his work on this project was invaluable in the completion of this paper.

A large arboreal Galois representation for a cubic postcritically finite polynomial

“…(See also [5,9,10].) More precisely, after counting elements of E n that fix a leaf of the tree T n , we have the following arithmetic application.…”

Odoni's conjecture for number fields

“…, s d−1 are transcendentals over F , then the generic monic polynomial G(x) = x d + s d−1 x d−1 + · · · + s 0 ∈ E[x] defined over the function field E satisfies Gal(E(G −n (0))/E) ∼ = [S d ] n . In [6], the second author showed that this result also holds for fields of characteristic p, except in the case p = d = 2. It follows from Hilbert's Irreducibility Theorem that if F = Q, or more generally if F is any Hilbertian field, then for any fixed n 0, there are infinitely many polynomials f (x) ∈ F (x) for which Gal(F (f −n (x 0 ))/F ) ∼ = [S d ] n .…”

Dynamical Galois groups of trinomials and Odoni's conjecture

“…We now set forth a criterion ensuring maximality of the extension . The ideas are inspired by the proof of [, Theorem 3.1]. Proposition Assume the hypotheses and notation of Proposition ; suppose, moreover, that for all , and that for all critical points of and , we have .…”