Bounds for preperiodic points for maps with good reduction

Let K be a number field and let ϕ in K(z) be a rational function of degree d≥2. Let S be the set of places of bad reduction for ϕ (including the archimedean places). Let Per(ϕ,K), PrePer(ϕ,K), and Tail(ϕ,K) be the set of K-rational periodic, preperiodic, and purely preperiodic points of ϕ, respectively. The present paper presents two main results. The first result is a bound for |PrePer(ϕ,K)| in terms of the number of places of bad reduction |S| and the degree d of the rational function ϕ. This bound significantly improves a previous bound given by J. Canci and L. Paladino. For the second result, assuming that |Per(ϕ,K)|≥4 (resp. |Tail(ϕ,K)|≥3), we prove bounds for |Tail(ϕ,K)| (resp. |Per(ϕ,K)|) that depend only on the number of places of bad reduction |S| (and not on the degree d). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when |Per(ϕ,K)|<4 (resp. |Tail(ϕ,K)|<3).