Waring numbers over finite commutative local rings

In this paper we study Waring numbers gR (k) for (R, m) a finite commutative local ring with identity and k E N with (k, |R|) = 1. We first relate the Waring number gR(k) with the diameter of the Cayley graphs GR(k) = Cay(R, UR(k)) and WR(k) = Cay(R, SR(k)) with UR (k) = {xk : x is an element of R*} and SR (k) = {xk : x is an element of Rx}, distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph GR(k) can be obtained by blowing-up the vertices of GFq(k) a number |m| of times, with independence sets the cosets of m, where q is the size of the residue field R/m. Then, by using the above blowing-up, we reduce the study of the Waring number gR (k) over the local ring R to the computation of the Waring number g(k, q) over the finite residue field R/m similar or equal to Fq. In this way, using known results for Waring numbers over finite fields, we obtain several explicit results for Waring numbers over finite commutative local rings with identity.(c) 2023 Elsevier B.V. All rights reserved.