Number systems and tilings over Laurent series

Abstract

AbstractLetbe a field and[x,y] the ring of polynomials in two variables over. Letf∈[x,y] and consider the residue class ringR:=[x,y]/f[x,y]. Our first aim is to study digit representations inR, i.e., we ask for whichfeach element ofRadmits a digit representation of the formd0+d1x+ ⋅ ⋅ ⋅ +dℓxℓwith digitsdi∈[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 ringS:=((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 ofxin their “x-ary” representation. The translates of the fundamental domain induce a tiling ofS. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose=qto be a finite field, these unions become finite.