A Lower Bound for the Height of a Rational Function at S-unit Points

By Pietro Corvaja, Umberto Zannier. 2003

83 citations recorded by the source corpus.

Abstract.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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a3ba9d75aaa6ad17148e6bf7cd28e7b8eaff562c

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Citations in the corpus

Generalized Greatest Common Divisors, Divisibility Sequences, and Vojta’s Conjecture for Blowups (2004, methodology): “For ease of exposition, we state their result over Q. Theorem 1 (Corvaja-Zannier [4]).”
Some cases of Vojta’s conjecture on integral points over function fields (2005, background): “This result, which is the function field analogue of Proposition 2 of [ CZ2 ], is of independent interest and admits further applications which will be developed elsewhere.”