Uniform bounds on periodic points of polynomials with good reduction
Authors
Isaac Rajagopal
Robin Zhang
Abstract
We establish effective bounds on the number of periodic points of degree-$d$ polynomials $φ$ 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 $\#\mathrm{Per}_K(φ) \leq d^{[K:\mathbb{Q}]}$.
