Sets of Lengths of Powers of a Variable

TL;DR

This paper characterizes the set of possible lengths of polynomials of the form x^n over certain Artinian local rings with nilpotent maximal ideals, providing insights into their factorization properties.

Contribution

It determines the set of lengths for polynomials x^n in R[x] where R is an Artinian local ring with m^2=0, a novel result in polynomial factorization over such rings.

Findings

Set of lengths explicitly characterized for x^n over R[x]

Identifies the structure of factorizations in rings with nilpotent maximal ideals

Provides a foundation for further study of polynomial factorizations in similar rings

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, m) is an Artinian local ring with m^2 = 0.