### Abstract

Let (Formula presented.) be a number field of degree k and let (Formula presented.) be an order in (Formula presented.). A generalized number system over(Formula presented.) (GNS for short) is a pair (Formula presented.) where (Formula presented.) is monic and (Formula presented.) is a complete residue system modulo p(0) containing 0. If each (Formula presented.) admits a representation of the form (Formula presented.) with (Formula presented.) and (Formula presented.) then the GNS (Formula presented.) is said to have the finiteness property. To a given fundamental domain (Formula presented.) of the action of (Formula presented.) on (Formula presented.) we associate a class (Formula presented.) of GNS whose digit sets (Formula presented.) are defined in terms of (Formula presented.) in a natural way. We are able to prove general results on the finiteness property of GNS in (Formula presented.) 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 (Formula presented.) we characterize the finiteness property of (Formula presented.) for fixed p and large (Formula presented.). Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

Original language | English |
---|---|

Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Monatshefte fur Mathematik |

DOIs | |

Publication status | Accepted/In press - May 18 2018 |

### Fingerprint

### Keywords

- Number field
- Number system
- Order
- Tiling

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Monatshefte fur Mathematik*, 1-24. https://doi.org/10.1007/s00605-018-1191-x

**Number systems over orders.** / Pethő, A.; Thuswaldner, Jörg.

Research output: Contribution to journal › Article

*Monatshefte fur Mathematik*, pp. 1-24. https://doi.org/10.1007/s00605-018-1191-x

}

TY - JOUR

T1 - Number systems over orders

AU - Pethő, A.

AU - Thuswaldner, Jörg

PY - 2018/5/18

Y1 - 2018/5/18

N2 - Let (Formula presented.) be a number field of degree k and let (Formula presented.) be an order in (Formula presented.). A generalized number system over(Formula presented.) (GNS for short) is a pair (Formula presented.) where (Formula presented.) is monic and (Formula presented.) is a complete residue system modulo p(0) containing 0. If each (Formula presented.) admits a representation of the form (Formula presented.) with (Formula presented.) and (Formula presented.) then the GNS (Formula presented.) is said to have the finiteness property. To a given fundamental domain (Formula presented.) of the action of (Formula presented.) on (Formula presented.) we associate a class (Formula presented.) of GNS whose digit sets (Formula presented.) are defined in terms of (Formula presented.) in a natural way. We are able to prove general results on the finiteness property of GNS in (Formula presented.) 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 (Formula presented.) we characterize the finiteness property of (Formula presented.) for fixed p and large (Formula presented.). Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

AB - Let (Formula presented.) be a number field of degree k and let (Formula presented.) be an order in (Formula presented.). A generalized number system over(Formula presented.) (GNS for short) is a pair (Formula presented.) where (Formula presented.) is monic and (Formula presented.) is a complete residue system modulo p(0) containing 0. If each (Formula presented.) admits a representation of the form (Formula presented.) with (Formula presented.) and (Formula presented.) then the GNS (Formula presented.) is said to have the finiteness property. To a given fundamental domain (Formula presented.) of the action of (Formula presented.) on (Formula presented.) we associate a class (Formula presented.) of GNS whose digit sets (Formula presented.) are defined in terms of (Formula presented.) in a natural way. We are able to prove general results on the finiteness property of GNS in (Formula presented.) 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 (Formula presented.) we characterize the finiteness property of (Formula presented.) for fixed p and large (Formula presented.). Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

KW - Number field

KW - Number system

KW - Order

KW - Tiling

UR - http://www.scopus.com/inward/record.url?scp=85047154745&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047154745&partnerID=8YFLogxK

U2 - 10.1007/s00605-018-1191-x

DO - 10.1007/s00605-018-1191-x

M3 - Article

AN - SCOPUS:85047154745

SP - 1

EP - 24

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

ER -