Number systems over general orders

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 $\mathcal{O}{\otimes }_{\mathbb{Z}}\mathbb{R}/\mathcal{O}$ and $p\in \mathcal{O}[X]$ a monic polynomial. For $\alpha \in \mathcal{O}$, define $p_{\alpha }(x):=p(x+\alpha )$ and ${\mathcal{D}}_{\mathcal{F},p(\alpha )}:=p(\alpha )\mathcal{F}\cap \mathcal{O}$. 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 $m$.

16 pages