Integral points in two-parameter orbits

TL;DR

This paper proves the finiteness and computability of pairs of integers (m,n) such that the m-th iterate of a point u is S-integral relative to the n-th iterate of a point w under a rational map, extending Silverman's work.

Contribution

It establishes a two-parameter analog of Silverman's result on integral points in orbits, using Runge's method and Vojta's theorem for effective and ineffective bounds.

Findings

The set of such (m,n) pairs is finite.

The finiteness is effectively computable.

The result generalizes Silverman's orbit integral points to two parameters.

Abstract

Let K be a number field, let f: P_1 --> P_1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w in P_1(K) are not preperiodic under f. We prove that the set of (m,n) in N^2 such that f^m(u) is S-integral relative to f^n(w) is finite and effectively computable. This may be thought of as a two-parameter analog of a result of Silverman on integral points in orbits of rational maps. This issue can be translated in terms of integral points on an open subset of P_1^2; then one can apply a modern version of the method of Runge, after increasing the number of components at infinity by iterating the rational map. Alternatively, an ineffective result comes from a well-known theorem of Vojta.