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 etale orders and GNS over $$\mathbb{Z}$$ . Their proofs depend on some general effective finiteness results of Evertse and Győry on monogenic etale orders.