Integral points in two-parameter orbits

Pietro Corvaja; Vijay Sookdeo; Thomas J. Tucker; Umberto Zannier

doi:10.1515/crelle-2013-0060

Article

Integral points in two-parameter orbits

Pietro Corvaja , Vijay Sookdeo , Thomas J. Tucker and Umberto Zannier

Published/Copyright: July 19, 2013

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2015 Issue 706

Abstract

Let K be a number field, let f:ℙ1→ℙ1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u,w∈ℙ1(K) are not preperiodic under f. We prove that the set of (m,n)∈ℕ2 such that f∘m(u) is S-integral relative to f∘n(w) is finite and effectively computable. This may be thought of as a two-parameter analog of a result of Silverman on integral points in orbits of rational maps. This issue can be translated in terms of integral points on an open subset of ℙ12; then one can apply a modern version of the method of Runge, after increasing the number of components at infinity by iterating the rational map. Alternatively, an ineffective result comes from a well-known theorem of Vojta.

Funding source: NSF

Award Identifier / Grant number: 1200749, 0854839

Funding source: ERC

Award Identifier / Grant number: Diophantine Problems

The authors thank Aaron Levin for many helpful conversations. They are also much indebted to the referee for suggesting a more conceptual proof of Theorem 4.1.

Received: 2012-3-8

Revised: 2013-6-17

Published Online: 2013-7-19

Published in Print: 2015-9-1

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https://doi.org/10.1515/crelle-2013-0060