Multiplicative dependence of rational values modulo approximate finitely generated groups

TL;DR

This paper proves finiteness results concerning the multiplicative dependence of rational values modulo sets close to finitely generated groups in number fields, extending understanding of algebraic relations in these contexts.

Contribution

It establishes new finiteness theorems for multiplicative dependence of rational function values modulo approximate finitely generated groups.

Findings

Finiteness of elements with multiplicative dependence under certain conditions

Results apply to rational functions over number fields

Extends previous work on algebraic dependence and height functions

Abstract

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are `close' (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field $K$. For example, we show that under some conditions on rational functions $f_1, \ldots, f_n\in K(X)$, there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets.