A Carlitz-von Staudt type theorem for finite rings

TL;DR

This paper generalizes the computation of power-sums over finite rings, unifying previous results and resolving a conjecture related to zeta values for matrix rings over finite commutative rings.

Contribution

It extends power-sum formulas to arbitrary finite rings, unifies prior work, and classifies translation-invariant polynomials over large classes of finite rings.

Findings

Unified power-sum formulas for all finite rings.

Resolved a conjecture on zeta values for matrix rings.

Classified translation-invariant polynomials over finite rings.

Abstract

We compute the $k$th power-sums (for all $k>0$) over an arbitrary finite unital ring $R$. This unifies and extends the work of Brawley, Carlitz, and Levine for matrix rings [Duke Math. J. 1974], with folklore results for finite fields and finite cyclic groups, and more general recent results of Grau and Oller-Marcen for commutative rings [Finite Fields Appl. 2017]. As an application, we resolve a conjecture by Fortuny Ayuso, Grau, Oller-Marcen, and Rua on zeta values for matrix rings over finite commutative rings [Internat. J. Algebra Comput. 2017]. We further recast our main result via zeta values over polynomial rings, and end by classifying the translation-invariant polynomials over a large class of finite commutative rings.