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 ∈ O [ x ] is monic and D ⊂ O is a complete residue system modulo p(0) containing 0. If each a ∈ O [ x ] admits a representation of the form a ≡ ∑ j = 0 ℓ - 1 d j x j ( mod p ) with ℓ ∈ N and d 0 , … , d ℓ - 1 ∈ 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 ∈ O [ x ] } 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 ± m ) , D F ) for fixed p and large 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.