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Arithmetic Dynamics of Diagonally Split Polynomial Maps
 Author(s): Nguyen, Khoa Dang
 Advisor(s): Scanlon, Thomas;
 Vojta, Paul
 et al.
Abstract
Let $K$ be a number field or the function field of a curve over an algebraically closed field of characteristic 0. Let $n\geq 2$, and let $f(X)\in K[X]$ be a polynomial of degree $d\geq 2$. We present two
arithmetic properties of the dynamics of the coordinatewise selfmap $\varphi=f\times\ldots\times f$ of $(\bP^1)^n$, namely the dynamical analogs of the Hasse principle and the BombieriMasserZannier height bound theorem. In particular, we prove that the Hasse principle holds when we intersect an orbit and a preperiodic subvariety, and that 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. A common crucial ingredient for the proof of these two properties is a recent classification of $\varphi$periodic subvarieties by MedvedevScanlon. We also present the problem of primitive prime divisors in dynamical sequences by IngramSilverman which is needed and closely related to the dynamical Hasse principle. Further questions on the bounded height result, and a possible generalization of the MedvedevScanlon classification are briefly given at the end.
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