Criteria for the integrality of $n$th roots of power series

TL;DR

This paper establishes necessary and sufficient conditions for when a power series with integer coefficients can be expressed as an nth power of another such series, expanding understanding of integrality criteria in power series.

Contribution

It provides a complete characterization of the integrality of nth roots of power series with integer coefficients, improving upon previous general criteria.

Findings

Derived necessary and sufficient conditions for nth roots of power series

Compared new criteria with Dieudonné and Dwork's integrality criterion

Enhanced understanding of power series root integrality conditions

Abstract

Heninger, Rains and Sloane raised the question of which power series with integer coefficients can be written as the $n$th power of another power series with integer coefficients and constant term $1$. We provide necessary and sufficient conditions, as well as compare with a general integrality criterion due to Dieudonn\'e and Dwork that can be applied to this question as well.