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

Ya-Qing Hu

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

Number Theory 2022-09-20 v3 Group Theory

Authors: Ya-Qing Hu

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 $n\ge1$ .

Keywords

Waring's problem locally nilpotent groups discrete Heisenberg groups polynomial maps polynomial sequences

Cite

@article{arxiv.2011.06683,
  title  = {Waring's Problem For Locally Nilpotent Groups: The Case of Discrete Heisenberg Groups},
  author = {Ya-Qing Hu},
  journal= {arXiv preprint arXiv:2011.06683},
  year   = {2022}
}

Comments

23 pages. Compared with the previous version, this one simplifies the main proof by adopting a technique that appears in Larsen and Nguyen's paper \cite{LN2019} and makes the main proof more readable

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

Abstract

Keywords

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