Polynomials assuming only local prime powers

TL;DR

This paper characterizes polynomials over local fields that map into their p-th powers, revealing that such polynomials form a broader class than just p-th powers of polynomials, with implications for number theory.

Contribution

It provides a new characterization of polynomials mapping local fields into p-th powers, expanding understanding beyond the class of p-th power polynomials.

Findings

The class of such polynomials is broader than p-th powers of polynomials.

A complete characterization of these polynomials over local fields.

Implications for the structure of polynomial mappings in number theory.

Abstract

We study the class of polynomials that map a local field (i.e., the completion of a number field at a non-Archimedean place) into the subset of its $p$-th powers, where $p$ is the residue characteristic of the field in question. We present a characterization of such polynomials and show that this class is always much broader than the class of $p$-th powers of polynomials.