Number systems and tilings over Laurent series By TOBIAS

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 of R admits a digit representation of the form d0 + d1x + · · · + dlx l with digits di ∈ F[y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. In a next step 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 := F((x, y))/f F((x, y)) 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 funda† The first author was supported by the FWF (Austrian Science Fund) research project SFB F1303 which is a part of the special research program “Numerical and Symbolic Scientific Computing”. ‡ The third author was supported by the FWF research projects P18079 and P20989. § The fourth author was supported by the FWF research project S9610 which is a part of the austrian research network “Analytic Combinatorics and Probabilistic Number Theory”. 2 Tobias Beck and Others mental domains of our digit systems turn out to be unions of boxes. If we choose F = Fq to be a finite field, these unions become finite.