Sums of Multivariate Polynomials in Finite Subgroups

Abstract

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 $I\subseteq \{1,\dots ,k\}$; then $$\sum x_1^{a_1}\cdots x_k^{a_k}\equiv \frac{\phi (p^s)}{\gcd (n,\phi (p^s))}(-1{)}^{k-1}(k-1)!\mathrm{mod}{p}^s,$$ where the summation is taken over all pairwise distinct $n$-th residues $x_1,\dots ,x_k$ modulo $p^s$ coprime with $p$.

10 pages, no figures. The article has been substantially rewritten and results have been improved. Old main result appears now as Corollary 4. Moreover, Theorem 2 has a proof which relies on a different argument