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2 results

  • Linear Forms in two m-adic Logarithms and Applications to Diophantine Problems

  • Yann Bugeaud
  • Journal:
    Compositio Mathematica / Volume 132 / Issue 2 / June 2002
    Published online by Cambridge University Press:
    04 December 2007, pp. 137-158
    Print publication:
    June 2002
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  • We give sharp, explicit estimates for linear forms in two logarithms, simultaneously for several non-Archimedean valuations. We present applications to explicit lower bounds for the fractional part of powers of rational numbers, and to the Diophantine equation (xn − 1)/(x − 1) = yq.

  • Quantitative Siegel's theorem for Galois coverings

  • YURI F. BILU
  • Journal:
    Compositio Mathematica / Volume 106 / Issue 2 / April 1997
    Published online by Cambridge University Press:
    04 December 2007, pp. 125-158
    Print publication:
    April 1997
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  • It is known that Siegel's theorem on integral points is effective for Galois coverings of the projective line. In this paper we obtain a quantitative version of this result, giving an explicit upper bound for the heights of S-integral K-rational points in terms of the number field K, the set of places S and the defining equation of the curve. Our main tools are Baker's theory of linear forms in logarithms and the quantitative Eisenstein theorem due to Schmidt, Dwork and van der Poorten.