A new way to tackle a conjecture of R\'emond

TL;DR

This paper introduces a novel approach to address Rémond's conjecture, which predicts a positive lower bound for the Weil height of elements outside a certain subgroup in a number field, related to the Zilber-Pink conjecture.

Contribution

The paper proposes a new method to investigate Rémond's conjecture, offering potential progress on bounding Weil heights in number fields.

Findings

Proposes a new approach to Rémond's conjecture.

Provides insights into Weil height bounds for algebraic numbers.

Connects the problem to broader conjectures in number theory.

Abstract

Let $\Gamma\subset \bar{\mathbb{Q}}^*$ be a finitely generated subgroup. Denote by $\Gamma_\mathrm{div}$ its division group. A recent conjecture due to R\'emond, related to the Zilber-Pink conjecture, predicts that the absolute logarithmic Weil height of an element of $\mathbb{Q}(\Gamma_\mathrm{div})^*\backslash \Gamma_{\mathrm{div}}$ is bounded from below by a positive constant depending only on $\Gamma$. In this paper, we propose a new way to tackle this problem.