Attaining potentially good reduction in arithmetic dynamics

Robert L. Benedetto

Published 2013-12-16, updated 2015-01-01Version 2

Let K be a non-archimedean field, and let f in K(z) be a rational function of degree d>1. If f has potentially good reduction, we give an upper bound, depending only on d, for the minimal degree of an extension L/K such that f is conjugate over L to a map of good reduction. In particular, if d=2 or d is greater than the residue characteristic of K, the bound is d+1. If K is discretely valued, we give examples to show that our bound is sharp.