Waring's Problem For Locally Nilpotent Groups: The Case of Discrete Heisenberg 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 $n\ge1$.