2017/08/16 by Pethő, Attila, Thuswaldner, Jörg
#11A63 #52C22 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1708.04800
Let \mathbbK be a number field of degree k and let O be an order in \mathbbK. A \emphgeneralized number system over O (GNS for short) is a pair (p,D) where p ∈ O[x] is monic and D\subsetO is a complete residue system modulo p(0) containing 0. If each a ∈ O[x] admits a representation of the form a ≡ ∑j =0ℓ-1 dj xj \pmodp with ℓ∈ℕ and d0,…, dℓ-1\inD then the GNS (p,D) is said to have the finiteness property. To a given fundamental domain F of the action of ℤk on ℝk we associate a class GF := \ (p, DF) : p ∈ O[x] \ of GNS whose digit sets DF are defined in terms of F in a natural way. We are able to prove general results on the finiteness property of GNS in 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 F we characterize the finiteness property of (p(x± m), DF) 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.