Number systems and tilings over Laurent series

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