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Uniformity in Polynomial Dynamics: Canonical Heights, Primitive Prime Divisors, and Galois Representations

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In this thesis, we study arithmetic phenomena exhibited by polynomial dynamical systems on the projective line. Specifically, given a number field $K$, we are interested in the arithmetic of orbits of points $\alpha\in K$ under polynomials $\phi\in K[z]$. Given such a polynomial $\phi$ of degree $d\ge2$, we prove a lower bound on the canonical height of any non-preperiodic $\alpha\in K$. This bound depends only on $d$, $K$, and the number of places of bad reduction of $\phi$. We then use this result to prove a uniform bound on the sizes of Zsigmondy sets associated to unicritical polynomials defined over a number field, assuming the $abc$-Conjecture. These theorems illustrate the principle that for a point $\alpha\in K$ having infinite forward orbit under $\phi$, we expect the iterates $\phi^n(\alpha)$ to exhibit arithmetically generic behavior as $n\to\infty$, in a way that is independent of the particular choices of $\alpha$ and $\phi$. Finally, we study the arithmetic of critical orbits in families of trinomials, and thereby prove several large image theorems for arboreal representations

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  • 04/09/2019
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