Scarcity of finite orbits for rational functions over a number field

TL;DR

This paper establishes new bounds on the number of rational preperiodic points for rational functions over number fields, improving previous bounds and linking the number of such points to the degree of the function and the existence of periodic points.

Contribution

It provides the first quadratic bound on the number of rational preperiodic points in terms of degree, and a linear bound under the assumption of a rational periodic point of period at least two.

Findings

Quadratic bound on rational preperiodic points in terms of degree d.

Linear bound on preperiodic points assuming a rational periodic point of period ≥ 2.

Improved bounds over previous results for rational functions over number fields.

Abstract

Let $\phi$ be a an endomorphism of degree $d\geq{2}$ of the projective line, defined over a number field $K$. Let $S$ be a finite set of places of $K$, including the archimedean places, such that $\phi$ has good reduction outside of $S$. The article presents two main results: the first result is a bound on the number of $K$-rational preperiodic points of $\phi$ in terms of the cardinality of the set $S$ and the degree $d$ of the endomorphism $\phi$. This bound is quadratic in terms of $d$ which is a significant improvement to all previous bounds on the number of preperiodic points in terms of the degree $d$. For the second result, if we assume that there is a $K$-rational periodic point of period at least two, then there exists a bound on the number of $K$-rational preperiodic points of $\phi$ that is linear in terms of the degree $d$.