Effective results for linear equations in two unknowns from a multiplicative division group

Abstract

We prove a new effective result for equations a1x1+a2x2 = 1 in (x1, x2) ∈ Γ, where a1, a2 are non-zero algebraic numbers and Γ is a finitely generated multiplicative subgroup of (Q)2. In our result, we give an explicit upper bound in terms of a1, a2 and Γ for the heights of the solutions (x1, x2). More generally, we prove effective results for solutions (x1, x2) in the division group of Γ and for solutions ‘close’ to this division group. Here, the solutions do not lie anymore in a given number field. Therefore, to achieve effectiveness we give for each solution (x1, x2), explicit upper bounds both for its height and for the degree of the number field it generates.