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

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.