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arXiv · 1201.1313

Integral points in two-parameter orbits

Pietro Corvaja·Vijay Sookdeo·Thomas J. Tucker·Umberto Zannier
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Abstract

Let K be a number field, let f: P_1 --> P_1 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 in P_1(K) are not preperiodic under f. We prove that the set of (m,n) in N^2 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 P_1^2; 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.

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Pietro Corvaja, Vijay Sookdeo, Thomas J. Tucker, Umberto Zannier. 2012-03-08. Integral points in two-parameter orbits. https://arxiv.org/abs/1201.1313

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