Number systems over orders

TL;DR

This paper studies generalized number systems over orders in number fields, establishing conditions for the finiteness property and linking these systems to power integral bases, with results depending on the topology of fundamental domains.

Contribution

It introduces a new abstract framework for analyzing the finiteness property of GNS over orders, connecting it to the topology of fundamental domains and power integral bases.

Findings

Characterization of the finiteness property for GNS in terms of the dominant condition.

Conditions under which GNS have the finiteness property for large shifts of the polynomial.

Established connections between GNS and power integral bases in number fields.

Abstract

Let $\mathbb{K}$ be a number field of degree $k$ and let $\mathcal{O}$ be an order in $\mathbb{K}$. A \emph{generalized number system over $\mathcal{O}$} (GNS for short) is a pair $(p,\mathcal{D})$ where $p \in \mathcal{O}[x]$ is monic and $\mathcal{D}\subset\mathcal{O}$ is a complete residue system modulo $p(0)$ containing $0$. If each $a \in \mathcal{O}[x]$ admits a representation of the form $a \equiv \sum_{j =0}^{\ell-1} d_j x^j \pmod{p}$ with $\ell\in\mathbb{N}$ and $d_0,\ldots, d_{\ell-1}\in\mathcal{D}$ then the GNS $(p,\mathcal{D})$ is said to have the \emph{finiteness property}. To a given fundamental domain $\mathcal{F}$ of the action of $\mathbb{Z}^k$ on $\mathbb{R}^k$ we associate a class $\mathcal{G}_\mathcal{F} := \{ (p, D_\mathcal{F}) \;:\; p \in \mathcal{O}[x] \}$ of GNS whose digit sets $D_\mathcal{F}$ are defined in terms of $\mathcal{F}$ in a natural way. We are able to prove general results on the finiteness property of GNS in $\mathcal{G}_\mathcal{F}$ by giving an abstract version of the well-known "dominant condition" on the absolute coefficient $p(0)$ of $p$. In particular, depending on mild conditions on the topology of $\mathcal{F}$ we characterize the finiteness property of $(p(x\pm m), D_\mathcal{F})$ for fixed $p$ and large $m\in\mathbb{N}$. Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.