On higher dimensional integrality and multiplicative dependence in semigroup algebraic dynamics

TL;DR

This paper investigates the relationship between integral points and multiplicative dependence in higher-dimensional semigroup orbits, linking non-density results to Vojta's conjecture.

Contribution

It extends previous results to higher dimensions, establishing that non-density of integral points implies sparsity of multiplicative dependence in semigroup orbits.

Findings

Non-density of integral points implies sparsity of multiplicative dependence.

The non-density hypothesis is implied by Vojta's conjecture.

Generalizes results of Northcott and Siegel to higher dimensions.

Abstract

We study multiplicative dependence of points in semigroup orbits in higher dimensions. More specifically, we show that the non-density of integral points in semigroup orbits implies sparsity of multiplicative dependence in orbits. This can be viewed as a semigroup dynamical and a higher dimensional version of recent results by B\'{e}rczes, Ostafe, Shparlinski and Silverman, which in turn can be viewed as a generalization of theorems of Northcott and Siegel. We also confirm that the non-density hypothesis of integral points in orbits is implied by Vojta's conjecture.