Uniform bounds on periodic points of polynomials with good reduction

We establish effective bounds on the number of periodic points of degree-$d$ polynomials $\phi$ defined over $p$-adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials $X^d+c$ with $c$ integral at some prime dividing $d$. As a consequence, we verify the uniform boundedness conjecture for this class of polynomials over number fields $K$, giving the explicit uniform bound $\mathrm{\#}{\mathrm{P}\mathrm{e}\mathrm{r}}_K(\phi )\le d^{[K:\mathbb{Q}]}$.