Height bound and preperiodic points for jointly regular families of rational maps

TL;DR

This paper extends a height inequality for jointly regular families of rational maps, providing bounds on the sum of heights of images of points, which has implications for understanding preperiodic points.

Contribution

It generalizes previous height inequalities from pairs to larger families of rational maps, improving bounds for jointly regular sets.

Findings

Established a new height inequality for jointly regular families of rational maps.

Extended previous results from pairs to larger sets of maps.

Provided bounds that relate heights of points and their images under multiple maps.

Abstract

Silverman proved a height inequality for jointly regular family of rational maps and the author improved it for jointly regular pairs. In this paper, we provide the same improvement for jointly regular family; if S is a jointly regular set of rational maps, then \sum_{f\in S} \dfrac{1}{\deg f} h\bigl(f(P) \bigr) > (1+ \dfrac{1}{r}) f(P) - C where r = \max_{f\in S} r(f).