Height coincidences in products of the projective line

TL;DR

This paper investigates hypersurfaces in products of projective lines containing generic small-height points, establishing strong relations between zero-height coordinates and advancing the dynamical Bogomolov conjecture.

Contribution

It proves that in certain hypersurfaces, zero-height in most coordinates implies all coordinates have zero height, aiding the resolution of the dynamical Bogomolov conjecture for split maps.

Findings

In hypersurfaces, zero height in most coordinates implies all coordinates have zero height.

Established strong coincidence relations between points with zero height coordinates.

Progress towards the dynamical Bogomolov conjecture for split maps.

Abstract

We consider hypersurfaces in $(\mathbb{P}^1)^n$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $n-1$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $n-1$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.