Uniform bounds on periodic points of polynomials with good reduction

TL;DR

This paper provides explicit uniform bounds on the number of periodic points for certain polynomials over p-adic and number fields, confirming a conjecture for a specific class of polynomials with good reduction.

Contribution

It establishes effective bounds on periodic points for polynomials with good reduction and verifies the uniform boundedness conjecture for unicritical polynomials over number fields.

Findings

Bound ^{[K:\u211aa9]} on periodic points

Verification of the uniform boundedness conjecture for unicritical polynomials

Applicable to polynomials with good reduction over number fields

Abstract

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 $\#\mathrm{Per}_K(\phi) \leq d^{[K:\mathbb{Q}]}$.