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Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 187–205 | Cite as

Number systems over general orders

  • J.-H. Evertse
  • K. GyőryEmail author
  • A. Pethő
  • J. M. Thuswaldner
Article
  • 18 Downloads

Abstract

Let \(\mathcal{O}\) be an order, that is a commutative ring with 1 whose additive structure is a free \(\mathbb{Z}\)-module of finite rank. A generalized number system (GNS for short) over \(\mathcal{O}\) is a pair \((p, \mathcal{D})\) where \(p \in \mathcal{O}[x]\) is monic with constant term p(0) not a zero divisor of \(\mathcal{O}\), and where \(\mathcal{D}\) is a complete residue system modulo p(0) in \(\mathcal{O}\) containing 0. We say that \((p, \mathcal{D})\) is a GNS over \(\mathcal{O}\) with the finiteness property if all elements of \(\mathcal{O}[x]/(p)\) have a representative in \(\mathcal{D}[x]\) (the polynomials with coefficients in \(\mathcal{D}\)). Our purpose is to extend several of the results from a previous paper of Pethő and Thuswaldner, where GNS over orders of number fields were considered. We prove that it is algorithmically decidable whether or not for a given order \(\mathcal{O}\) and GNS \((p, \mathcal{D})\) over \(\mathcal{O}\), the pair \((p, \mathcal{D})\) admits the finiteness property. This is closely related to work of Vince on matrix number systems.

Let \(\mathcal{F}\) be a fundamental domain for \(\mathcal{O}{\otimes}_\mathbb{Z} \mathbb{R}/\mathcal{O}\, {\rm and}\, p \in \mathcal{O}[X]\) a monic polynomial. For \(\alpha \in \mathcal{O}\), define \(p_{\alpha}(x) := p(x+\alpha) {\rm and} \mathcal{D}_{\mathcal{F},p(\alpha)} := p(\alpha)\mathcal{F} \bigcap \mathcal{O}\). Under mild conditions we show that the pairs \((p_{\alpha},\mathcal{D}_{\mathcal{F},p(\alpha)})\) are GNS over \(\mathcal{O}\) with finiteness property provided \(\alpha \in \mathcal{O}\) in some sense approximates a sufficiently large positive rational integer. In the opposite direction we prove under different conditions that \((p_{-m},\mathcal{D}_{\mathcal{F},p(-m)})\) does not have the finiteness property for each large enough positive rational integer m.

We obtain important relations between power integral bases of étale orders and GNS over \(\mathbb{Z}\). Their proofs depend on some general effective finiteness results of Evertse and Győry on monogenic étale orders.

Mathematics Subject Classification

11A63 11R04 

Key words and phrases

number system order tiling 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  • J.-H. Evertse
    • 1
  • K. Győry
    • 2
    Email author
  • A. Pethő
    • 3
    • 4
  • J. M. Thuswaldner
    • 5
  1. 1.Mathematical InstituteLeiden UniversityLeidenThe Netherlands
  2. 2.Institute of MathematicsUniversity of DebrecenDebrecenHungary
  3. 3.Department of Computer ScienceUniversity of DebrecenDebrecenHungary
  4. 4.Faculty of ScienceUniversity of OstravaOstravaCzech Republic
  5. 5.Chair of Mathematics and StatisticsUniversity of LeobenLeobenAustria

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