Sums of Multivariate Polynomials in Finite Subgroups

Let R be a commutative ring, f ∈ R[X1,⋯,Xk] 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 Σ f(x1,⋯,xk), where the summation is taken over all pairwise distinct x1,⋯,xk G. In particular, let ps be a power of an odd prime, n a positive integer coprime with p - 1, and a1,⋯,ak integers such that φ(ps) divides a1 + ⋯ + ak and p - 1 does not divide Σi∈Iai for all non-empty proper subsets I ⊆ {1,⋯,k}; then ∑x1a1⋯xkak ≡ φ(ps)/gcd(n,φ(ps))(-1)k-1(k - 1)!modps, where the summation is taken over all pairwise distinct nth residues x1,⋯,xk modulo ps coprime with p.