A lower bound for the height of a rational function at S-unit points

Let a; b be given, multiplicatively independent positive integers and let epsilon > 0. In a recent paper jointly with Y. Bugeaud we proved the upper bound expd(epsilon n) for g.c.d. (a(n) - 1; b(n) - 1); shortly afterwards we generalized this to the estimate g. c. d. (u - 1; v - 1)< max(\u\,\v\)(epsilon) for multiplicatively independent S-units u; v is an element of Z. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g. c. d. to pairs of rational functions other than {u - 1, v - 1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G(m)(2).