Counting points on diagonal equations over Galois rings GR(p²,p^2r)

Abstract

Let $R=GR(p^2,p^{2r})$ be a Galois ring and $N(a_1{x_1}^{k_1}+\cdots +a_n{x}_n^{k_n}=b)$ denote the number of solutions of diagonal equations $a_1{x_1}^{k_1}+\cdots +a_n{x}_n^{k_n}=b$ in R, where $a_1,\cdots ,a_n\in R\backslash \{0\}$, $x_1,\cdots ,x_n,b\in R$. In this paper, we will express N by Jacobi sums in the finite field ${\mathbb{F}}_{p^r}$. In particular, the precise number of solutions can be obtained when $k_1=k_2=\cdots =k_n=2$, but in general some estimates can be satisfied.