Sums of Multivariate Polynomials in Finite Subgroups

TL;DR

This paper evaluates sums of multivariate polynomials over finite subgroups of units in rings, providing explicit formulas under specific algebraic conditions, with applications to sums over residues modulo prime powers.

Contribution

It introduces a method to compute sums of multivariate polynomials over finite subgroups of units, extending previous results to more general algebraic settings.

Findings

Explicit sum formulas for polynomials over finite subgroups

Conditions under which the sums are non-zero

Application to sums over residues modulo prime powers

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 $p$.