Twisted Thue equations with multiple exponents in fixed number fields

Abstract

Let $K$ be a number field of degree $d\ge 3$ and fix $s$ multiplicatively independent algebraic integers ${\gamma }_1,\dots ,{\gamma }_s\in K^{\ast }$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel's conjecture. We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-γ_1^{t_1}\cdotsγ_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y,(t_1,\dots ,t_s))$ with $xy\ne 0$ and $\mathbb{Q}\left({\gamma }_1^{t_1}\cdots {\gamma }_s^{t_s}\right)=K$, all of which are effectively computable.