On relative fields of definition for log pairs, Vojta's height inequalities and asymptotic coordinate size dynamics

Abstract

We build on the perspective of the works \cite{Grieve:Noytaptim:fwd:orbits}, \cite{Matsuzawa:2023}, \cite{Grieve:qualitative:subspace}, \cite{Grieve:chow:approx}, \cite{Grieve:Divisorial:Instab:Vojta} (and others) and study the dynamical arithmetic complexity of rational points in projective varieties. Our main results make progress towards the attractive problem of asymptotic complexity of coordinate size dynamics in the sense formulated by Matsuzawa, in \cite[Question 1.1.2]{Matsuzawa:2023}, and building on earlier work of Silverman \cite{Silverman:1993}. A key tool to our approach here is a novel formulation of conjectural Vojta type inequalities for log canonical pairs and with respect to finite extensions of number fields. Among other features, these conjectured Diophantine arithmetic height inequalities raise the question of existence of log resolutions with respect to finite extensions of number fields which is another novel concept which we formulate in precise terms here and also which is of an independent interest.