Linear equations in variables which lie in a multiplicative group

Lot K be a field of characteristic 0 and lot n be a natural number. Let Gamma be a subgroup of the multiplicative group (K*)(n) of finite rank r. Given a(1), ..., a(n) is an element of K* write A(a(1), ..., a(n), Gamma) for the number of solutions x = (x(1), ..., x(n)) is an element of Gamma of the equation 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), Gamma) which depends only on the dimension n and on the rank r.