Waring numbers over finite commutative local rings

PODESTÁ, RICARDO A.; VIDELA, DENIS E.

DISCRETE MATHEMATICS

ELSEVIER SCIENCE BV

Año: 2023 vol. 346

0012-365X

In this paper we study Waring numbers for a finite commutative local ring with identity and with . We first relate the Waring number with the diameter of the Cayley graphs and with ⁎ and , distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph can be obtained by blowing-up the vertices of a number of times, with independence sets the cosets of , where q is the size of the residue field . Then, by using the above blowing-up, we reduce the study of the Waring number over the local ring R to the computation of the Waring number over the finite residue field . 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.