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

TL;DR

This paper extends Waring's problem to the ring of integers modulo p^k, analyzing the solvability of specific k-th power sum equations and providing bounds on solutions for the case s=2.

Contribution

It introduces new bounds on the number of solutions for the sum of two k-th powers in rac{p^k}{}, extending classical Waring's problem to modular rings.

Findings

Derived an upper bound for solutions when s=2

Analyzed solvability conditions for k-th power sum equations

Extended Waring's problem to rac{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 $s=2$.