Inhomogeneous Thue Equations

Let α be an algebraic integer of degree n ≥ 3, \(K={\mathbb Q}(\alpha )\), and let \(0\neq m\in {\mathbb Z}\). In some applications for index form equations in sextic and octic fields (cf. Sects. 11.2.1, 11.2.2, and 14.2.3) we shall need to solve equations of type $$\displaystyle N_{K/{\mathbb Q}}(x+\alpha y +\lambda )=m \;\;\; \mathrm {in} \;\;\; x,y\in {\mathbb Z}, \lambda \in {\mathbb Z}_K, $$ where we assume that \(\overline {|\lambda |}overline {|\lambda |}\) denotes the size of λ, that is the maximum absolute value of its conjugates). Sprindžuk (J Number Theory 6:481–486, 1974) considered equations of this type. This equation might also be referred to as inhomogeneous Thue equation. Using Baker’s method he gave effective upper bounds for the solutions of the above equation. The variables x, y are called dominating variables, while λ is called non-dominating variable. In Gaal (Math Comput 51:359–373, 1988) we gave a numerical method of solving (4.1), which we briefly explain below. It is interesting to compare these arguments with the relevant arguments for Thue equations and to see how the standard estimates can be modified to cover the inhomogeneous case.