Density of Unicritical Polynomials over Local Fields with Rational Periodic Points

Abstract

We prove an asymptotic formula for the density of parameters $c\in {\mathbb{Z}}_p$ for which the polynomial $x^d+c$ admits a ${\mathbb{Q}}_p$-rational $n$-cycle. In the cases where $d=2$ and $p\ne 2,3$, we compute these densities explicitly for $n=1,2,$ and $3$.

18 pages