The equation x+y=1 in finitely generated groups

has not more than 3× 7d+2s solutions. Since s ≥ d/2 this implies that (2) has at most 3× 74s solutions. We can apply this result to equation (1). However, the estimate will depend on the degree of the field containing H, and on s, the number of places for which the elements of H have non-trivial valuation. Note that for fixed r the number s may have arbitrarily large values. We shall be interested in bounds which depend only on r. The first such uniform result for a general subgroup G of (C∗)2 was given in [5]. There the bound