2024/07/11 by Nathan Grieve, Grieve, Nathan, Chatchai Noytaptim +1
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2407.08614
openalex publication_date 2024/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Working over a base number field \KK, we study the attractive question of Zariski non-density for (D,S)-integral points in Of(x) the forward f-orbit of a rational point x ∈ X(\KK). Here, f \colon X → X is a regular surjective self-map for X a geometrically irreducible projective variety over \KK. Given a non-zero and effective f-quasi-polarizable Cartier divisor D on X and defined over \KK, our main result gives a sufficient condition, that is formulated in terms of the f-dynamics of D, for non-Zariski density of certain dynamically defined subsets of Of(x). For the case of (D,S)-integral points, this result gives a sufficient condition for non-Zariski density of integral points in Of(x). Our approach expands on that of Yasufuku, \citeYasufuku:2015, building on earlier work of Silverman \citeSilverman:1993. Our main result gives an unconditional form of the main results of loc.~cit.; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in \citeRu:Vojta:2016 and expanded upon in \citeGrieve:points:bounded:degree and \citeGrieve:qualitative:subspace.