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

Abstract

Effective results for linear equations in two unknowns from a multiplicative division group by Attila Bérczes (Debrecen), Jan-Hendrik Evertse (Leiden) and Kálmán Győry (Debrecen)1. Introduction.In the literature there are various effective results on S-unit equations in two unknowns.In our paper we work out effective results in a quantitative form for the more general equationwhere a 1 , a 2 ∈ Q * and Γ is an arbitrary finitely generated subgroup of positive rank of the multiplicative group (Qordinatewise multiplication (see Theorems 2.1 and 2.2).Such more general results can be used to improve upon existing effective bounds on the solutions of discriminant equations and certain decomposable form equations.These will be worked out in a forthcoming work.In fact, in the present paper we prove even more general effective results for equations of the shape (1.1) with solutions (x 1 , x 2 ) from a larger group, namely the division group Γ2 ) ∈ Γ }, and even with solutions (x 1 , x 2 ) "very close" to Γ .To our knowledge, these are the first effective results of this kind.Our results give an effective upper bound for both the height of a solution (x 1 , x 2 ) and the degree of the field Q(x 1 , x 2 ); see Theorems 2.3 and 2.5 and Corollary 2.4.In the proofs of these theorems we utilize Theorem 2.1 (on (1.1) with solutions from Γ ), as well as a result of Beukers and Zagier [2], which asserts that (1.1) has at most two solutions (x 1 , x 2 ) ∈ (Q * ) 2 with very small height.The hard core of the proofs of our results mentioned above is a new effective lower bound for |1 -αξ| v , where α is a fixed element from a given algebraic number field K, v is a place of K, and the unknown ξ is taken from 2000

Locations

• Acta Arithmetica - View - PDF