Number systems over general orders

TL;DR

This paper extends the theory of generalized number systems over orders in algebraic structures, providing algorithms to decide finiteness properties and exploring conditions under which these systems have the finiteness property.

Contribution

It generalizes previous results to broader orders, introduces an algorithm for finiteness property decision, and analyzes the impact of polynomial shifts on GNS finiteness.

Findings

Decidability of the finiteness property for GNS over orders.

Conditions under which polynomial shifts yield GNS with finiteness property.

Results on non-finiteness for large polynomial shifts.

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 Peth\H{o} 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 $m$.