Linear equations with unknowns from a multiplicative group in a function field

Let k be an algebraically closed field of characteristic 0, let K/k be a transcendental extension of arbitrary transcendence degree and let G be a multiplicative subgroup of (K^)^n such that (k^)^n is contained in G, and G/(k^)^n has finite rank r. We consider linear equations a1x1+…+anxn=1 () with fixed non-zero coefficients a1,…,an from K, and with unknowns (x1,…,xn) from the group G. Such a solution is called degenerate if there is a subset of a1x1,…,anxn whose sum equals 0. Two solutions (x1,…,xn), (y1,…,yn) are said to belong to the same (k^)^n-coset if there are c1,…,cn in k^ such that y1=c1x1,…,yn=cnxn. We show that the non-degenerate solutions of () lie in at most 1+C(3,2)^r+C(4,2)^r+…+C(n+1,2)^r (k^)^n-cosets, where C(a,b) denotes the binomial coefficient a choose b.