2015/01/15 by Yu Yasufuku · 4 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Combinatorics #Log-polar coordinates #Mathematical analysis #Mathematical physics #Mathematics #Orthogonal coordinates #Physics #math.DS #math.NT #msc:11J97 #msc:37P15 #msc:37P55
paper · pdf · doi:10.1007/s00209-015-1406-y
published in Mathematische Zeitschrift 279(3-4), 1121-1141 (Springer Science+Business Media) · 22 pages, to appear in Mathematische Zeitschrift
arxiv created 2015/01/15 · arxiv updated 2015/01/16 · openalex publication_date 2015/01/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We give a generalization to higher dimensions of Silverman's result on finiteness of integer points in orbits. Assuming Vojta's conjecture, we prove a sufficient condition for morphisms on PN so that (S,D)-integral points in each orbit are Zariski-non-dense. This condition is geometric, and for dimension 1 it corresponds precisely to Silverman's hypothesis that the second iterate of the map is not a polynomial. In fact, we will prove a more precise formulation comparing local heights outside S to the global height. For hyperplanes, this amounts to comparing logarithmic sizes of the coordinates, generalizing Silverman's precise version in dimension 1. We also discuss a variant where we can conclude that integral points in orbits are finite, rather than just Zariski-non-dense. Further, we show unconditional results and examples, using Schmidt's subspace theorem and known cases of Lang--Vojta conjecture. We end with some extensions to the case of rational maps and to the case when the arithmetic of the orbit under one map is controlled by the geometric properties of another. We include many explicit examples to illustrate different behaviors of integral points in orbits in higher dimensions.