Density of Unicritical Polynomials over Local Fields with Rational Periodic Points

Anna M. Dietrich

Density of Unicritical Polynomials over Local Fields with Rational Periodic Points

Number Theory 2026-07-17 v1

Authors: Anna M. Dietrich

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 $p\neq 2,3$ , we compute these densities explicitly for $n=1,2,$ and $3$ .

Keywords

unicritical polynomials local fields rational periodic points density p-adic dynamics

Cite

@article{arxiv.2607.16349,
  title  = {Density of Unicritical Polynomials over Local Fields with Rational Periodic Points},
  author = {Anna M. Dietrich},
  journal= {arXiv preprint arXiv:2607.16349},
  year   = {2026}
}

Comments

18 pages