Waring’s problem for locally nilpotent groups: The case of discrete Heisenberg groups

Abstract

Kamke (Kamke, 1921) solved an analog of Waring’s problem with $n$th power monomials replaced by integer-valued polynomials. Larsen and Nguyen (Larsen and Nguyen, 2019) 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 a previous work (Hu, 2024) to solve an analog of Waring’s problem for the general discrete Heisenberg group $H_{2n+1}\left(Z\right)$ for each integer $n\ge 1$.