Eventually stable rational functions

TL;DR

This paper investigates the stability of polynomial factorizations related to iterated rational functions over fields, proposing a conjecture for global fields and proving it under specific conditions, with applications to S-integral points.

Contribution

It introduces the concept of eventual stability for pairs (phi, alpha), formulates a conjecture over global fields, and proves it for maps with good reduction and bijective action on residue extensions.

Findings

Proves the conjecture for maps with good reduction and bijective residue action.

Establishes finiteness of S-integral points in backwards orbits for these maps.

Provides characterizations of eventual stability through finiteness conditions.

Abstract

For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n denotes the nth iterate of phi. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (phi, alpha) is called eventually stable over K. We conjecture that (phi, alpha) is eventually stable over K when K is any global field and alpha any point not periodic under phi (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when K has a discrete valuation for which (1) phi has good reduction and (2) phi acts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of S-integral points in backwards orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.