Number Systems and Tilings over Laurent Series

Let F be a field and F[x, y] the ring of polynomials in two variables over F. Let f ∈ F[x, y] and consider the residue class ring R := F[x, y]/fF[x, y]. Our first aim is to study digit representations in R, i.e., we ask for which f each element r ∈ R admits a digit representation of the form d0 + d1x + · · · + d`x with digits di ∈ F[y] satisfying degy di < degy f . These digit systems are motivated by the well-known notion of canonical number system. In a next step we enlarge the ring in order to allow for representations including negative powers of the “base” x. More precisely we define and characterize digit representations for the ring F((x−1, y−1))/fF((x−1, y−1)) and give easy to handle criteria for finiteness and periodicity. Finally, we attach fundamental domains to our number systems. The fundamental domain of a number system is the set of all numbers having only negative powers of x in their “x-ary” representation. Interestingly, the fundamental domains of our number systems turn out to be unions of boxes. If we choose F = Fq to be a finite field, these unions become finite.