Sesión Álgebra y GeometríaWaring numbers over finite commutative local rings
Ricardo A. Podestá
Universidad Nacional de Córdoba (FaMAF, CIEM-CONICET), Argentina - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
In this talk, based on the joint work [1], we study Waring numbers $g_R(k)$ for $(R,\frak m)$ a finite commutative local ring with identity and $k \in \mathbf{N}$ with $(k,|R|)=1$. We first relate the Waring number $g_R(k)$ with the diameter of the Cayley graphs $G_R(k)=Cay(R,U_R(k))$ and $W_R(k)=Cay(R,S_R(k))$ with $U_R(k)=\{x^k : x\in R^*\}$ and $S_R(k)=\{x^k : x\in R^\times\}$, distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph $G_R(k)$ can be obtained by blowing-up the vertices of $G_{\mathbf{F}_{q}}(k)$ a number $|\frak{m}|$ of times, with independence sets the cosets of $\frak{m}$, 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 $g_R(k)$ over the local ring $R$ to the computation of the Waring number $g(k,q)$ over the finite residue field $R/\frak m \simeq \mathbf{F}_q$. 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.
Trabajo en conjunto con: Denis E. Videla (Universidad Nacional de Córdoba, FaMAF, CIEM-CONICET).
Referencias
[1] Ricardo A. Podestá, Denis E. Videla. \textit{Waring numbers over finite commutative local rings}, Discrete Mathematics \textbf{346:10}, 10/2023, Art ID 113567, 22 págs., \url{https://doi.org/10.1016/j.disc.2023.113567}, (arXiv:2212.1239, 12/2022).