Number systems over orders.

Let $K$ be a number field of degree k and let $O$ be an order in $K$ . A generalized number system over $O$ (GNS for short) is a pair $(p,D)$ where $p\in O\left[x\right]$ is monic and $D\subset O$ is a complete residue system modulo p(0) containing 0. If each $a\in O\left[x\right]$ admits a representation of the form $a\equiv {\sum }_{j=0}^{\ell -1}{d}_j{x}^j(modp)$ with $\ell \in N$ and $d_0,\dots ,d_{\ell -1}\in D$ then the GNS $(p,D)$ is said to have the finiteness property. To a given fundamental domain $F$ of the action of $Z^k$ on $R^k$ we associate a class $G_F:=\{(p,D_F):p\in O\left[x\right]\}$ of GNS whose digit sets $D_F$ are defined in terms of $F$ in a natural way. We are able to prove general results on the finiteness property of GNS in $G_F$ 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 $F$ we characterize the finiteness property of $(p(x\pm m),D_F)$ for fixed p and large $m\in N$ . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.