A Carlitz–von Staudt Type Theorem for Finite Rings

We compute the kth power-sum polynomials (for k > 0) over an arbitrary finite ring R, obtained by summing the kth powers of (T + r) for r ∈ R. For R non-commutative, this extends the work of Brawley–Carlitz–Levine [Duke Math. J. 41], and resolves a conjecture by Fortuny Ayuso, Grau, Oller-Marcén, and Rúa (2015). For R commutative, our results bring together two classical programs in the literature: von Staudt–Clausen type results on computing zeta values in finite rings [J. reine angew. Math. 21]; and computing power-sum polynomials over finite fields, which arises out of the work of Carlitz on zeta functions [Duke Math. J. 5,7]. Our proof in this case crucially uses symmetric function theory. Along the way, we also classify the translation-invariant polynomials over a wide class of finite commutative rings.