Number systems over orders

Attila Pethő, Jörg Thuswaldner

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Let K be a number field of degree k and let O be an order in K. A generalized number system overO (GNS for short) is a pair (p, D) where p∈ O[x] is monic and D⊂ O is a complete residue system modulo p(0) containing 0. If each a∈ O[x] admits a representation of the form a≡∑j=0ℓ-1djxj(modp) with ℓ∈ N and d0, … , d - 1∈ D then the GNS (p, D) is said to have the finiteness property. To a given fundamental domain F of the action of Zk on Rk we associate a class GF:={(p,DF):p∈O[x]} of GNS whose digit sets DF are defined in terms of F in a natural way. We are able to prove general results on the finiteness property of GNS in GF 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± m) , DF) for fixed p and large m∈ N. Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

Original languageEnglish
Pages (from-to)681-704
Number of pages24
JournalMonatshefte fur Mathematik
Volume187
Issue number4
DOIs
Publication statusPublished - Dec 1 2018

Keywords

  • Number field
  • Number system
  • Order
  • Tiling

ASJC Scopus subject areas

  • Mathematics(all)

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