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Uniform bounds for the number of solutions to Yn = f(X)

Published online by Cambridge University Press:  24 October 2008

J.-H. Evertse  and
J. H. Silverman
J.-H. Evertse
Affiliation:
Centre of Mathematics and Computer Science, 1009 AB Amsterdam, The Netherlands
J. H. Silverman
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
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Extract

Let K be an algebraic number field and f(X) ∈ K[X]. The Diophantine problem of describing the solutions to equations of the form

has attracted considerable interest over the past 60 years. Siegel [12], [13] was the first to show that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K. LeVeque[7] proved the following, more explicit, result. Let

where a ∈ K* and αl,…,αk are distinct and algebraic over K. Then (+) has only finitely many integral solutions unless (nl,…,nk) is a permutation of one of the n-tuples

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 2 , September 1986 , pp. 237 - 248
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

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