Sets of lengths of powers of a variable

Abstract

A positive integer $k$ is a length of a polynomial if that polynomial factors into a product of $k$ irreducible polynomials. We find the set of lengths of polynomials of the form $x^n$ in $R[x]$, where $(R, \mathfrak{m} )$ is an Artinian local ring with $\mathfrak{m} ^2=0$.