Waring's Problem For Locally Nilpotent Groups: The Case of Discrete Heisenberg Groups

TL;DR

This paper extends Waring's problem to discrete Heisenberg groups using polynomial map theory, providing new insights into the representation of group elements as products of polynomial sequences.

Contribution

It applies polynomial map theory to solve an analog of Waring's problem specifically for discrete Heisenberg groups, a novel setting in this context.

Findings

Solved Waring's problem for discrete Heisenberg groups

Established polynomial map techniques in nilpotent group context

Extended previous algebraic group results to specific nilpotent groups

Abstract

Kamke \cite{Kamke1921} solved an analog of Waring's problem with $n$th powers replaced by integer-valued polynomials. Larsen and Nguyen \cite{LN2019} explored the view of algebraic groups as a natural setting for Waring's problem. This paper applies the theory of polynomial maps and polynomial sequences in locally nilpotent groups developed in previous work \cite{Hu2020} to solve an analog of Waring's problem for the general discrete Heisenberg groups $H_{2n+1}(\mathbb{Z})$ for any integer $n\ge1$.