Waring's problem for unipotent algebraic groups

TL;DR

This paper explores an analogue of Waring's problem within algebraic groups, proving that for unipotent groups over certain fields, every element can be expressed as a bounded product of images of a morphism from the affine line.

Contribution

It establishes the first results on Waring's problem for unipotent algebraic groups over characteristic zero and totally imaginary number fields.

Findings

Every element of unipotent groups over suitable fields is a bounded product of morphism images.

The results extend to integral points over rings of integers in totally imaginary number fields.

Provides a new perspective on expressing group elements as products in algebraic geometry.

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.