Waring numbers over finite commutative local rings

Abstract

In this paper we study Waring numbers gR(k) for (R,m) a finite commutative local ring with identity and k∈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∈R⁎} and SR(k)={xk:x∈R×}, 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≃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.