2012/01/05 by Pietro Corvaja, Corvaja, Pietro, Vijay Sookdeo +5
Mathematics · #37P55 #FOS: Mathematics #Number Theory (math.NT) #Primary: 14G25 #Secondary: 37F10 #math.NT #msc:14G25 #msc:37F10 #msc:37P55
paper · pdf · doi:10.48550/arxiv.1201.1313
17 pages; minor revisions made
arxiv created 2012/03/08 · arxiv updated 2012/03/09
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 in P1(K) are not preperiodic under f. We prove that the set of (m,n) in N2 such that fm(u) is S-integral relative to fn(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.