Heights and totally $p$-adic numbers

TL;DR

This paper investigates the behavior of canonical height functions on totally p-adic fields, establishing a gap in their values and applying this to answer a longstanding question about polynomial dynamics over number fields.

Contribution

It proves a height gap for rational maps on totally p-adic fields and resolves Narkiewicz's 1963 question on polynomial invariance in number field compositums.

Findings

Established a positive lower bound for canonical heights on totally p-adic fields.

Proved no infinite invariant subset exists in the union of all number fields of bounded degree for certain polynomials.

Abstract

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the maximal totally $p$-adic field if the map $f$ has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset $X$ in the compositum of all number fields of degree at most $d$ such that $f(X)=X$ for some non-linear polynomial $f$. This answers a question of W. Narkiewicz from 1963.