Digit systems in polynomial rings over finite fields

In the present paper we define a new kind of digit system in polynomial rings over finite fields. Let be a finite field and be a polynomial in two indeterminates. Furthermore, let . Then is a digit system with base p(x,y) if each element q of the quotient has a finite representation of the shape q=d0(x)+d1(x)y+ +dh(x)yhwith (0 i h). We give a complete characterization of all polynomials p(x,y) which give rise to such a digit system and establish some arithmetical properties of the set of all such polynomials. There are striking analogies of these digit systems to the well-known canonical number systems defined in for a polynomial .