Number systems over orders

Attila Pethő; Jörg Thuswaldner

doi:10.1007/s00605-018-1191-x

Number systems over orders

Mon Hefte Math. 2018;187(4):681-704. doi: 10.1007/s00605-018-1191-x. Epub 2018 May 18.

Authors

Attila Pethő^{1

2}, Jörg Thuswaldner³

Affiliations

¹ 1Department of Computer Science, University of Debrecen, P.O. Box 12, Debrecen, 4010 Hungary.
² 2Faculty of Science, University of Ostrava, Dvořákova 7, 70103 Ostrava, Czech Republic.
³ 3Chair of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, 8700 Leoben, Austria.

Abstract

Let $K$ be a number field of degree k and let $O$ be an order in $K$ . A generalized number system over $O$ (GNS for short) is a pair $(p, D)$ where $p \in O [x]$ is monic and $D \subset O$ is a complete residue system modulo p(0) containing 0. If each $a \in O [x]$ admits a representation of the form $a \equiv \sum_{j = 0}^{ℓ - 1} d_{j} x^{j} (mod p)$ with $ℓ \in N$ and $d_{0}, \dots, d_{ℓ - 1} \in D$ then the GNS $(p, D)$ is said to have the finiteness property. To a given fundamental domain $F$ of the action of $Z^{k}$ on $R^{k}$ we associate a class $G_{F} : = {(p, D_{F}) : p \in O [x]}$ of GNS whose digit sets $D_{F}$ are defined in terms of $F$ in a natural way. We are able to prove general results on the finiteness property of GNS in $G_{F}$ 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 $F$ we characterize the finiteness property of $(p (x \pm m), D_{F})$ for fixed p and large $m \in N$ . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

Keywords: Number field; Number system; Order; Tiling.