Uniform bounds on periodic points of polynomials with good reduction

Abstract

We establish effective bounds on the number of periodic points of degree-$d$ polynomials $\ddot{I}•$ 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 $\#{\mathrm{Per}}_K(\ddot{I}•)\le d^{[K:\mathbb{Q}]}$.

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