2 2 A pr 2 00 4 A lower bound for the height of a rational function at S - unit points Pietro
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چکیده
Abstract. Let a, b be given multiplicatively independent positive integers and let ǫ > 0. In a recent paper written jointly also with Y. Bugeaud we proved the upper bound exp(ǫn) for gcd(a − 1, b − 1); shortly afterwards we generalized this to the estimate gcd(u− 1, v − 1) < max(|u|, |v|), 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 Gm.
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تاریخ انتشار 2008