Counting points of bounded height in monoid orbits

TL;DR

This paper provides bounds on the number of rational points of bounded height within monoid orbits generated by endomorphisms on projective space, with refined estimates for free monoids and generic rational functions.

Contribution

It establishes upper and lower bounds for points of bounded height in monoid orbits, including refined bounds for free monoids and generic rational functions.

Findings

Upper bounds on points of bounded height in monoid orbits.

Refined lower bounds for free monoids.

Most rational functions satisfy these bounds.

Abstract

Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.