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1. Density results for PCF polynomials

Assume $f(z)\in \mathbb{Q}[z]$ is PCF (orbit of critical points is finite) and some reasonable starting point $\alpha\in\mathbb{Q}.$
    1. Problem 1.1.

      [Xander Faber] Describe \[ N(x)=\#\{p\leq x : p|f^n(\alpha) \text { for some } n\} \] for $p$ prime as $x\rightarrow \infty.$
        1. Remark. There are results for non PCF maps.
            • Remark. The guess here is that $N(x)=o(\pi(x)).$
                • Problem 1.2.

                  [Joe Silverman] What is the growth rate of $N(x)$ for PCF and non PCF maps?
                    • Problem 1.3.

                      If $f(z)\in\mathbb{F}_q(T)$, what happens then? What happens with other fields?

                          Cite this as: AimPL: The Galois theory of orbits in arithmetic dynamics, available at http://aimpl.org/galarithdyn.