Some arithmetic dynamics of diagonally split polynomial maps. (English) Zbl 1391.37073

Nguyen, Khoa

Int. Math. Res. Not. 2015, No. 5, 1159-1199 (2015).

Summary: 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 \((f,\ldots,f):(\mathbb P^1_K)^n\to (\mathbb P^1_K)^n\), namely the dynamical analogs 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.

MSC:

37P05	Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P15	Dynamical systems over global ground fields
11G35	Varieties over global fields

Keywords:

Hasse principle; Bombieri-Masser-Zannier height bound; split polynomial map; periodic hypersurface