A quantitative estimate for quasi-integral points in orbits

TL;DR

This paper provides an explicit upper bound on the number of quasi-S-integral points in the forward orbits of rational functions of degree at least 2, under certain non-polynomial second iterate conditions.

Contribution

It offers a quantitative estimate for the number of quasi-integral points in orbits, extending previous finiteness results with explicit bounds.

Findings

Established an explicit upper bound for quasi-S-integral points in orbits

Extended finiteness results to quantitative estimates

Applicable to rational functions with non-polynomial second iterate

Abstract

Let f(z) be a rational function of degree at least 2 with coefficients in a number field K, and assume that the second iterate f^2(z) of f(z) is not a polynomial. The second author previously proved that for any b in K, the forward orbit O_f(b) contains only finitely many quasi-S-integral points. In this note we give an explicit upper bound for the number of such points.