Scarcity of cycles for rational functions over a number field

Bounds for preperiodic points for maps with good reduction

“…While the work described in this paper was being carried out, Canci and Vishkautsan [8] proved a bound for | Per(φ, K)|, just assuming that φ has good reduction outside S. Their bound on | Per(φ, K)| is roughly of the order of d2 16|S| + 2 2187|S| where d ≥ 2 is the degree of φ.…”

“…Using equations (6) and (7) we get x = u1x2 y2x1−y1x2 − u2x1 y2x1−y1x2 and y = u1y2 y2x1−y1x2 − u2y1 y2x1−y1x2 . Then by (8)…”

“…Then | PrePer(φ c , Q)| ≤ 9. B. Hutz and P. Ingram [10] have shown that Poonen's conjecture holds when the numerator and denominator of c don't exceed 10 8 .…”

Cycles for rational maps over global function fields with one prime of bad reduction

“…The next result is of a different nature: in this case we consider points with coordinates in a global field K. Let S be a finite set of primes of K; by putting together local information about p-adic distances for all primes outside S we establish a connection between points with certain properties and solutions of unit equations. The n-points lemmas are central tools in the work of Canci, Troncoso and Vishkautsan over number fields (see [5] and [6]) where they are applied in combination with bounds for the number of solutions of the resulting unit equations. The three points lemma we present here is a qualitative version of [5,Corollary 4.2]; its proof can be adapted from that of [5,Lemma 4.1] but we include it here for the reader's convenience.…”

Scarcity of finite orbits for rational functions over a number field

“…A natural relaxation of the uniform boundedness conjecture is to restrict our study to families of rational functions given in terms of good reduction. A rational map φ : P 1 → P 1 of degree d ≥ 2 defined over a number field K is said to have good reduction at a non zero prime p of K if φ can be written as φ = [F (X, Y ) : G(X, Y )] where F, G ∈ R p [X, Y ] are homogeneous polynomials of degree d, such that the resultant of F and G is a p-unit, where R p is the localization of the ring of integers of K at p. The map φ is said to have bad reduction at a prime p of K if φ does not have good reduction at p. For a fixed finite set S of places of K containing all the archimedean ones, we say that φ has good reduction outside of S if it has good reduction at each place p / ∈ S. In the special case of rational functions φ : P 1 → P 1 , there are several results giving a uniform bound on the number of periodic/preperiodic points of φ depending on the cardinality of a finite set of places S, which includes all archimedean places, together with the constants [K : Q] and deg(φ), under the assumption that φ has good reduction outside of S (e.g., [Nar89,MS94,Ben07,Can07,Can10,CP16,CV,Tro17]).…”