Linear equations in variables which lie in a multiplicative group

Let K be a field of characteristic 0 and let n be a natural number. Let Γ be a subgroup of the multiplicative group (K∗)n of finite rank r. Given a1,..., an ∈ K∗ write A(a1,..., an,Γ) for the number of solutions x = (x1,..., xn) ∈ Γ of the equation a1x1 +... + anxn = 1, such that no proper subsum of a1x1 +... + anxn vanishes. We derive an explicit upper bound for A(a1,..., an,Γ) which depends only on the dimension n and on the rank r.