An Extension of Waring's Problem in $\mathbb{Z}_{p^k}$

Abstract

In this paper, we will investigate the solvability of the equation $x_1^k + x_2^k + \ldots + x_s^k = n$, $n\in \mathbb{Z}_{p^k}$, $x_1,...,x_s\in \mathcal{A}$, $\mathcal{A}\subseteq \mathbb{Z}_{p^k}$. We will give a upper bound of the number of solutions for $s=2$.

We feel that the proofs of two main Lemmas 2.7 and 4.7 seems not impeccable, and unconvincing more or less