Waring numbers over finite commutative local rings

Number of cliques of Paley-type graphs over finite commutative local rings

“…Corollary 2.4 [24]. Let (R, m) be a finite commutative local ring with m = |m| and residue field R/m…”

“…x ∈ R * } with R a finite commutative ring with identity; when (2). Some structural properties of G R (k) were studied in the case where R is local in [24] (see also [6] for R = Z p α ). In general, notice that G R (k) is directed, moreover U R (k) is a symmetric set if and only if…”

“…In this work, we generalize the results obtained by Bhowmik and Barman over R = Z p α with k = 2, to all finite commutative local rings R with k = 2, 3, 4 (see Theorems 3.1 and 4.1). More precisely we study the cliques of G R (k) over finite commutative local rings with identity R, where (k, |R|) = 1, through the decomposition of G R (k) in terms of G Fq (k) recently found in [24]. We will reduce the computation of the number of K (G R (k)) over finite commutative local rings (R, m) to the problem of computing K (Γ(k, q)) where Γ(k, q) is as in (2).…”

“…Remark 5. By taking R = Z p α in item (a) of Theorem 4.1, we have that β = α−1 and q = p prime, in this case we have that (24) K 4 (G Z p α (2)) = p 4α−3 (p − 1)((p − 9) 2 − 16y 2 ) 2 9 • 3 ,…”