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- W2963273515 abstract "Let $K$ be a number field or the function field of a curve over an algebraically closed field of characteristic 0. Let $ngeq 2$, and let $f(X)in K[X]$ be a polynomial of degree $dgeq 2$. We present twoarithmetic properties of the dynamics of the coordinate-wise self-map $varphi=ftimesldotstimes f$ of $(bP^1)^n$, 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 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 Medvedev-Scanlon. We also present the problem of primitive prime divisors in dynamical sequences by Ingram-Silverman which is needed and closely related to the dynamical Hasse principle. Further questions on the bounded height result, and a possible generalization of the Medvedev-Scanlon classification are briefly given at the end." @default.
- W2963273515 created "2019-07-30" @default.
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- W2963273515 date "2013-11-13" @default.
- W2963273515 modified "2023-09-28" @default.
- W2963273515 title "Some Arithmetic Dynamics of Diagonally Split Polynomial Maps" @default.
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- W2963273515 doi "https://doi.org/10.1093/imrn/rnt240" @default.
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