Some Arithmetic Dynamics of Diagonally Split Polynomial Maps

TL;DR

This paper explores the arithmetic dynamics of diagonal polynomial maps over number and function fields, establishing analogues of classical principles and height bounds using recent theorems.

Contribution

It applies Medvedev-Scanlon's theorem to prove dynamical versions of the Hasse principle and height bounds for polynomial maps in higher dimensions.

Findings

Hasse principle holds for orbit and preperiodic subvariety intersections.

Points on a curve intersecting periodic hypersurfaces have bounded heights unless special conditions.

Provides new dynamical analogues of classical arithmetic principles.

Abstract

Let $n\geq 2$, and let $f$ be a polynomial of degree at least 2 with coefficients in a number field or a characteristic 0 function field $K$. We present two arithmetic applications of a recent theorem of Medvedev-Scanlon to the dynamics of the map $(f,...,f):\ (\bP^1_{K})^n\longrightarrow (\bP^1_K)^n$, namely the dynamical analogues of the Hasse principle and the Bombieri-Masser-Zannier height bound theorem. In particular, we prove that the Hasse principle holds when we intersect an orbit and a preperiodic subvariety, and that points in the intersection of a curve with the union of all periodic hypersurfaces have bounded heights unless that curve is vertical or contained in a periodic hypersurface.