Sums of Multivariate Polynomials in Finite Subgroups

Abstract: Let $R$ be a commutative ring, $f \in R[X_1,\ldots,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\sum f(x_1,\ldots,x_k)$, where the summation is taken over all pairwise distinct $x_1,\ldots,x_k \in G$. In particular, let $p^s$ be a power of an odd prime, $n$ a positive integer coprime with $p-1$, and $a_1,\ldots,a_k$ integers such that $\varphi(p^s)$ divides $a_1+\cdots+a_k$ and $p-1$ does not divide $\sum_{i \in I}a_i$ for all non-empty proper subsets $I\subseteq {1,\ldots,k}$; then $$ \sum x_1^{a_1}\cdots x_k^{a_k} \equiv \frac{\varphi(p^s)}{\mathrm{gcd}(n,\varphi(p^s))}(-1)^{k-1}(k-1)! \,\,\bmod{p^s}, $$ where the summation is taken over all pairwise distinct $n$-th residues $x_1,\ldots,x_k$ modulo $p^s$ coprime with $p$.