  Let K be a field of characteristic 0 and let n be a natural number. Let Gamma
be a subgroup of the multiplicative group $(K^\ast)^n$ of finite rank r. Given
$A_2,...,a_n\in K^\ast$ write $A(a_1,...,a_n,\Gamma)$ for the number of
solutions x=(x_1,...,x_n)\in \Gamma$ of the equation a_1x_1+...+a_nx_n=1$, such
that no proper subsum of $a_1x_1+...+a_nx_n$ 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.
