Number systems over orders

Abstract

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