Waring Numbers of Ramified $p$-adic Rings

Let $p$ be prime and $e,k$ be positive integers, and let $R={\mathbb{Z}}_p[\sqrt[e]{p}]$. We calculate the Waring numbers $g_R(k)$ for many values of $p,e$, and $k$, and investigate how the Waring numbers for $p=2$ change as $e$ and $k$ vary.