Let $${\mathbb {K}}$$K be a number field of degree k and let $${\mathcal {O}}$$O be an order in $${\mathbb {K}}$$K. A generalized number system over$${\mathcal {O}}$$O (GNS for short) is a pair $$(p,{\mathcal {D}})$$(p,D) where $$p \in {\mathcal {O}}[x]$$p∈O[x] is monic and $${\mathcal {D}}\subset {\mathcal {O}}$$D⊂O is a complete residue system modulo p(0) containing 0. If each $$a \in {\mathcal {O}}[x]$$a∈O[x] admits a representation of the form $$a \equiv \sum _{j =0}^{\ell -1} d_j x^j \pmod {p}$$a≡∑j=0ℓ-1djxj(modp) with $$\ell \in {\mathbb {N}}$$ℓ∈N and $$d_0,\ldots , d_{\ell -1}\in {\mathcal {D}}$$d0,…,dℓ-1∈D then the GNS $$(p,{\mathcal {D}})$$(p,D) is said to have the finiteness property. To a given fundamental domain $${\mathcal {F}}$$F of the action of $${\mathbb {Z}}^k$$Zk on $${\mathbb {R}}^k$$Rk we associate a class $${\mathcal {G}}_{\mathcal {F}} := \{ (p, D_{\mathcal {F}}) \;:\; p \in {\mathcal {O}}[x] \}$$GF:={(p,DF):p∈O[x]} of GNS whose digit sets $$D_{\mathcal {F}}$$DF are defined in terms of $${\mathcal {F}}$$F in a natural way. We are able to prove general results on the finiteness property of GNS in $${\mathcal {G}}_{\mathcal {F}}$$GF by giving an abstract version of the well-known “dominant condition” on the absolute coefficient p(0) of p. In particular, depending on mild conditions on the topology of $${\mathcal {F}}$$F we characterize the finiteness property of $$(p(x\pm m), D_{\mathcal {F}})$$(p(x±m),DF) for fixed p and large $$m\in {\mathbb {N}}$$m∈N. Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

中文翻译：

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订单数制

令 $${\mathbb {K}}$$K 是度数为 k 的数域，令 $${\mathcal {O}}$$O 是 $${\mathbb {K}}$$K 中的一个阶. $${\mathcal {O}}$$O（简称 GNS）上的广义数系统是一对 $$(p,{\mathcal {D}})$$(p,D) 其中 $$p \在 {\mathcal {O}}[x]$$p∈O[x] 是 monic 并且 $${\mathcal {D}}\subset {\mathcal {O}}$$D⊂O 是一个完整的残差系统模 p(0) 包含 0。如果每个 $$a \in {\mathcal {O}}[x]$$a∈O[x] 承认 $$a \equiv \sum _{j = 0}^{\ell -1} d_j x^j \pmod {p}$$a≡∑j=0ℓ-1djxj(modp) 与 $$\ell \in {\mathbb {N}}$$ℓ∈N和 $$d_0,\ldots , d_{\ell -1}\in {\mathcal {D}}$$d0,...,dℓ-1∈D 那么 GNS $$(p,{\mathcal {D}} )$$(p,D) 被称为具有有限性。$${\mathbb {Z}}^k$$Zk 对 $${\mathbb {R}}^k$$Rk 的作用的给定基本域 $${\mathcal {F}}$$F我们关联一个类 $${\mathcal {G}}_{\mathcal {F}} := \{ (p, D_{\mathcal {F}}) \;:\; p \in {\mathcal {O}}[x] \}$$GF:={(p,DF):p∈O[x]} 的数字集合 $$D_{\mathcal {F}}$ 的 GNS $DF 以自然的方式根据 $${\mathcal {F}}$$F 定义。我们能够通过给出一个众所周知的“显性条件”的抽象版本来证明 $${\mathcal {G}}_{\mathcal {F}}$$GF 中 GNS 有限性的一般结果。 p 的绝对系数 p(0)。特别是，根据 $${\mathcal {F}}$F 拓扑的温和条件，我们刻画了 $$(p(x\pm m), D_{\mathcal {F}})$ 的有限性属性$(p(x±m),DF) 对于固定 p 和大 $$m\in {\mathbb {N}}$$m∈N。使用我们的新理论，