Monatshefte für Mathematik

, Volume 144, Issue 3, pp 203–224

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

  • Pietro Corvaja
  • Umberto Zannier

DOI: 10.1007/s00605-004-0273-0

Cite this article as:
Corvaja, P. & Zannier, U. Mh Math (2005) 144: 203. doi:10.1007/s00605-004-0273-0


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,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.

2000 Mathematics Subject Classifications: 11J25 
Key words: Lower bounds for the height, Subspace Theorem, Linear tori 

Copyright information

© Springer-Verlag/Wien 2004

Authors and Affiliations

  • Pietro Corvaja
    • 1
  • Umberto Zannier
    • 2
  1. 1.University of UdineItaly
  2. 2.Scuola Normale SuperiorePisaItaly

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