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 : A 1 → G and ask whether every element of G(K) is the product of a bounded number of elements f (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(O) can be written as a product of a bounded number of elements of f (O). We prove this is the case when G is unipotent and O is the ring of integers of a totally imaginary number field.

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

“…Recently, Larsen and Nguyen [LN19] explored the idea of algebraic groups as a natural setting for Waring's problem. The work on the polynomial-valued, vector-valued and certain matrix-valued variants of Waring's problem can naturally fit into this framework.…”

“…Hence, they have to choose between either working over a totally imaginary number ring or doing the easier Waring problem on a general number ring. Moreover, a natural question has been raised in [LN19]: whether, for unipotent groups over general number rings O, one can characterize the set which ought to be expressible as a bounded product of images of the morphism f : A 1 → U k over the ring O.…”