Relative Thue Equations

Let M ⊂ K be algebraic number fields, and let K = M(α) with an algebraic integer α. Let \(0\neq \mu \in {\mathbb Z}_M\). Consider the relative Thue equation $$\displaystyle N_{K/M}(X-\alpha Y)=\mu \;\;\; \mathrm {in} \;\;\; X,Y\in {\mathbb Z}_M. $$ Equations of this type were first considered in effective form by Kotov and Sprindzuk (Dokl Akad Nauk BSSR 17:393–395, 477, 1973). This equation is a direct analogue of ( 3.1) in the relative case, when the ground ring is \({\mathbb Z}_M\) instead of \({\mathbb Z}\). The equation given in this form has only finitely many solutions. Relative Thue equations are often considered in the form $$\displaystyle N_{K/M}(X-\alpha Y)=\eta \mu \;\;\; \mathrm {in}\;\;\; X,Y\in {\mathbb Z}_M, $$ where η is an unknown unit in M. In this case the solutions are determined up to a unit factor of M.