Waring Numbers of Ramified $p$-adic Rings

TL;DR

This paper computes the Waring numbers for certain ramified p-adic rings, revealing how these numbers vary with parameters p, e, and k, especially focusing on the case p=2.

Contribution

It provides explicit calculations of Waring numbers for ramified p-adic rings and analyzes their behavior as parameters change, a novel extension in number theory.

Findings

Calculated Waring numbers for various p, e, k values

Identified patterns in Waring numbers for p=2 as e and k vary

Extended understanding of additive representations in ramified p-adic rings

Abstract

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.