Twisted Thue equations with multiple exponents in fixed number fields

TL;DR

This paper studies a class of twisted Thue equations over number fields involving multiple exponents and proves finiteness and effective computability of solutions under certain independence conditions.

Contribution

It establishes finiteness and effective bounds for solutions to twisted Thue equations with multiple exponents in fixed number fields, extending previous results.

Findings

Finiteness of solutions under specified conditions

Solutions are effectively computable

Applicable to equations with multiple multiplicatively independent algebraic integers

Abstract

Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $\gamma_1, \dots, \gamma_s \in K^*$ 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 \[ \left|N_{K/\mathbb{Q}}\left(X-\gamma_1^{t_1}\cdots\gamma_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( \gamma_1^{t_1}\cdots \gamma_s^{t_s} \right) = K$, all of which are effectively computable.