arXiv 1810.097102018-10math.NT

一般序上的数制

Number systems over general orders

Jan-Hendrik Evertse, Kálmán Győry, Attila Pethő, Jörg M. Thuswaldner

备注：: 16 pages

摘要：

设$\mathcal{O}$是一个序，即具有$1$的交换环，其加法结构是有限秩的自由$\mathbb{Z}$ -模。$\mathcal{O}$上的广义数制（简称GNS）是一对$(p,\mathcal{D} )$，其中$p\in\mathcal{O}[x]$与常数项$p(0)$（不是$\mathcal{O}$的零因子）是一元的，并且其中$\mathcal{D}$是包含$0$的$\mathcal{O}$中模$p(0)$的完全剩余系.如果$\mathcal{O}[x]/(p)$的所有元素在$\mathcal{D}[x]$中都有一个代表（系数在$\mathcal{D}$中的多项式），我们称$(p,\mathcal{D})$是$\mathcal{O}$上的GNS，且具有有限性.我们的目的是延长PethIgnand Thuswalnut，其中GNS的顺序数域被认为是从以前的文件的结果。我们证明了它是算法判定是否为一个给定的顺序$\mathcal{O}$和GNS $(p,\mathcal{D})$在$\mathcal{O}$，对$(p,\mathcal{D})$承认有限性。这是密切相关的工作文斯矩阵编号系统。设$\mathcal{F}$是$\mathcal{O} \!\otimes_{\mathbb{Z}}\! \mathbb{R}/\mathcal{O}$的基本整环，$p\in \mathcal{O}[X]$是一元多项式.对于$α\in\mathcal{O}$，定义PDMATHBNB和$\mathcal{D}_{\mathcal{F} ,p(α)}:= p(α)\mathcal{F}\cap\mathcal{O}$。在较弱的条件下，证明了当$α\in\mathcal{O}$在某种意义上逼近一个充分大的正有理数时，对$(p_α,\mathcal{D}_{\mathcal{F},p(α)}\,)$是$\mathcal{O}$上具有有限性的GNS.在相反的方向，我们证明了在不同的条件下，$(p_{-m},\mathcal{D}_{\mathcal{F} ,p(-m)}\,)$不具有有限性的每一个足够大的正有理整数$m$。

英文摘要：

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 $α\in\mathcal{O}$, define $p_α(x):=p(x+α)$ and $\mathcal{D}_{\mathcal{F} ,p(α)}:= p(α)\mathcal{F}\cap\mathcal{O}$. Under mild conditions we show that the pairs $(p_α,\mathcal{D}_{\mathcal{F},p(α)}\,)$ are GNS over $\mathcal{O}$ with finiteness property provided $α\in\mathcal{O}$ 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 $m$.

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