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The Diophantine equation $ \alpha _{1}^{x_{1}}\cdots \alpha _{n}^{x_{n}} = f(x_{1},\dots ,x_{n})$. II


Authors: P. Corvaja, W. M. Schmidt and U. Zannier
Journal: Trans. Amer. Math. Soc. 362 (2010), 2115-2123
MSC (2000): Primary 11D61, 11D45; Secondary 11R18
DOI: https://doi.org/10.1090/S0002-9947-09-05012-0
Published electronically: November 18, 2009
MathSciNet review: 2574889
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Abstract | References | Similar Articles | Additional Information

Abstract: We will deal with the equation of the title where $ \alpha _{1},\dots ,\alpha _{n}$ are multiplicatively independent complex numbers and $ f$ is a polynomial. We will give a bound for the number of solutions which depends only on $ n$ and the degree of $ f$. Two further results which play a rôle in the proof are of independent interest.


References [Enhancements On Off] (What's this?)

  • 1. J.-H. Evertse. The number of solutions of linear equations in roots of unity. Acta Arith. 89 (1999), 45-51. MR 1692199 (2000e:11033)
  • 2. G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. 3rd ed. Clarendon Press, Oxford (1954). MR 0067125 (16:673c)
  • 3. M. Laurent. Équations exponentielles polynômes et suites récurrentes linéaires. Astérisque 147-148 (1987), 343-344.
  • 4. M. Laurent. Équations exponentielles polynômes et suites récurrentes linéaires, II. J. Number Theory 31 (1989), 24-53. MR 978098 (90b:11023)
  • 5. A. Schinzel and W. M. Schmidt. Powers of Roots in Linear Spaces. Journal of Number Theory (to appear).
  • 6. H. P. Schlickewei and W. M. Schmidt. The Number of Solutions of Polynomial-Exponential Equations. Compositio Math. 120 (2000), 193-225. MR 1739179 (2001b:11022)
  • 7. W. M. Schmidt. The zero multiplicity of linear recurrence sequences. Acta Math. 182 (1999), 243-282. MR 1710183 (2000j:11043)
  • 8. W. M. Schmidt. The Diophantine Equation $ \alpha _{1}^{x_{1}} \cdots \alpha _{n}^{x_{n}} = f(x_{1},\dots ,x_{n})$. Analytic Number Theory. Essays in Honour of Klaus Roth, 414-420. Cambridge University Press (2009). MR 2508660

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Additional Information

P. Corvaja
Affiliation: Department of Mathematics and Informatics, University of Udine, Via delle Scienze 206, 33100 Udine, Italy

W. M. Schmidt
Affiliation: Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309-0395

U. Zannier
Affiliation: Department of Mathematics, Scuola Normale Superiore, Piazza de Cavalier, 56100 Pisa, Italy

DOI: https://doi.org/10.1090/S0002-9947-09-05012-0
Received by editor(s): May 13, 2008
Published electronically: November 18, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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