Linear equations in variables which lie in a multiplicative group

J. -H. Evertse; H. P. Schlickewei; W. M. Schmidt

paper

Linear equations in variables which lie in a multiplicative group

arXiv:math/0409604

Abstract

Let K be a field of characteristic 0 and let n be a natural number. Let Gamma be a subgroup of the multiplicative group $(K^{*})^{n}$ of finite rank r. Given $A_{2}, ..., a_{n} \in K^{*}$ write $A (a_{1}, ..., a_{n}, Γ)$ for the number of solutions x=(x_1,...,x_n)\in Γ $o f t h ee q u a t i o n a_{1} x_{1} + ... + a_{n} x_{n} = 1$ , such that no proper subsum of $a_{1} x_{1} + ... + a_{n} x_{n}$ vanishes. We derive an explicit upper bound for $A (a_{1}, ..., a_{n}, Γ)$ which depends only on the dimension n and on the rank r.