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 $\mathcal{O}{\otimes}_\mathbb{Z} \mathbb{R}/\mathcal{O}\, {\rm and}\, p \in \mathcal{O}[X]$ 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.