2022/12/23 by Ricardo A. Podestá, Podestá, Ricardo A., Denis E. Videla +1 · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2212.12396
openalex publication_date 2022/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study Waring numbers gR(k) for (R,\frak 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 G_\mathbbFq(k) a number |\frakm| of times, with independence sets the cosets of \frakm, where q is the size of the residue field R/\frak 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/\frak m ≃ \mathbbFq. 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.