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 ε>0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(εn) for g.c.d.(an−1, bn−1); shortly afterwards we generalized this to the estimate g.c.d.(u−1,v−1)<max(∣u∣,∣v∣)ε for multiplicatively independent S-units u,v∈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 Gm2.