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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. Add remark

      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)).


                • Add remark

                  Problem 1.2.

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


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                      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.