Power sums over commutative and unitary rings

TL;DR

This paper generalizes the calculation of power sums over finite rings, extending known results from specific cases like finite fields and integer rings to more general commutative unital rings, with applications to polynomial quotient rings.

Contribution

It provides a unified method to compute power sums over a broad class of finite commutative rings, including polynomial quotient rings.

Findings

Power sums are explicitly computed for finite commutative rings.

Results extend classical formulas from finite fields and integers to general rings.

Applications include analysis of polynomial quotient rings.

Abstract

In this paper we compute the sum of the $k$-th powers over any finite commutative unital rings, thus generalizing known results for finite fields, the rings of integers modulo $n$ or the ring of Gaussian integers modulo $n$. As an application we focus on quotient rings of the form $\mathbb{Z}/n\mathbb{Z}[x]/(f(x))$ for any polynomial $f\in\mathbb{Z}[x]$