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

Abstract

In this paper, we will investigate the solvability of the equation $x_1^k+x_2^k+\dots +x_s^k=n$, $n\in {\mathbb{Z}}_{p^k}$, $x_1,...,x_s\in \mathcal{A}$, $\mathcal{A}\subseteq {\mathbb{Z}}_{p^k}$. 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