Counting points on diagonal equations over Galois rings GR(p2,p2r)

Abstract

Let R=GR(p2,p2r) be a Galois ring and N(a1x1k1+⋯+anxnkn=b) denote the number of solutions of diagonal equations a1x1k1+⋯+anxnkn=b in R, where a1,⋯,an∈R\\{0}, x1,⋯,xn,b∈R. In this paper, we will express N by Jacobi sums in the finite field Fpr. In particular, the precise number of solutions can be obtained when k1=k2=⋯=kn=2, but in general some estimates can be satisfied.