A Lower Bound for the Height of a Rational Function at S-unit Points |
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Authors: | Pietro Corvaja Umberto Zannier |
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Affiliation: | (1) University of Udine, Italy;(2) Scuola Normale Superiore, Pisa, Italy |
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Abstract: | Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(an–1, bn–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)u,v) for multiplicatively independent S-units u,vZ. 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. |
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Keywords: | 2000 Mathematics Subject Classifications: 11J25 |
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