Sums of Multivariate Polynomials in Finite Subgroups

Let $R$ be a commutative ring, $f\in R[X_1,\dots ,X_k]$ 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 $\sum f(x_1,\dots ,x_k)$, where the summation is taken over all pairwise distinct $x_1,\dots ,x_k\in G$. In particular, let $p^s$ be a power of an odd prime, $n$ a positive integer coprime with $p-1$, and $a_1,\dots ,a_k$ integers such that $\phi (p^s)$ 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$$ \sum x_1^{a_1}\cdots x_k^{a_k} \equiv \frac{\varphi(p^s)}{$\mathrm{g}\mathrm{c}\mathrm{d}$(n,\varphi(p^s))}(-1)^{k-1}(k-1)! ,,\bmod{p^s}, $$wherethesummationistakenoverallpairwisedistinct$n$-thresidues$x_1,\ldots,x_k$modulo$p^s$coprimewith$p$.