Waring's problem for unipotent algebraic groups

Abstract

In this paper, we formulate an analogue of Waring's problem for an algebraic group $G$. At the field level we consider a morphism of varieties $f\colon \mathbb{A}^1\to G$ and ask whether every element of $G(K)$ is the product of a bounded number of elements $f(\mathbb{A}^1(K)) = f(K)$. We give an affirmative answer when $G$ is unipotent and $K$ is a characteristic zero field which is not formally real. The idea is the same at the integral level, except one must work with schemes, and the question is whether every element in a finite index subgroup of $G(\mathcal{O})$ can be written as a product of a bounded number of elements of $f(\mathcal{O})$. We prove this is the case when $G$ is unipotent and $\mathcal{O}$ is the ring of integers of a totally imaginary number field.