%0 Journal Article
%T Sums of Multivariate Polynomials in Finite Subgroups
%A Paolo Leonetti
%A Andrea Marino
%J Mathematics
%D 2014
%I arXiv
%X Given a finite subgroup $G$ of the group of units of a commutative unital ring $R$ and a multivariate polynomial $f$ in $R[X_1,\ldots,X_k]$, we evaluate the sum of the $f(x_1,\ldots,x_k)$ for all choices of pairwise distinct $x_1,\ldots,x_k$ in $G$ whenever the subgroup $G$ satisfies a minimax constraint, which always holds if $R$ is a field. In particular, let $p^m$ be a power of an odd prime, $n$ a positive integer, and $a_1,\ldots,a_k$ integers with sum divisible by $\varphi(p^m)$ such that $\mathrm{gcd}(a_{i_1}+\cdots+a_{i_j},p(p-1))$ is smaller than $(p-1)/\mathrm{gcd}(n,\varphi(p^m))$ for all non-empty proper subsets $\{i_1,\ldots,i_j\}$ of $\{1,\ldots,k\}$; then the following congruence holds $$\sum x_1^{a_1}\cdots x_k^{a_k} \equiv \frac{\varphi(p^m)}{\mathrm{gcd}(n,\varphi(p^m))}(-1)^{k-1}(k-1)! \hspace{1mm}\pmod{p^m}, %(-1)^{k}(k-1)!p^{m-1} \bmod{p^m},$$ where the summation is taken over all pairwise distinct $1\le x_1,\ldots,x_k\le p^m$ such that each $x_i$ is a $n$-th residue modulo $p^m$ coprime with $p$.
%U http://arxiv.org/abs/1411.2269v2