Lengths, periods and expansions of elements written with respect to a digit system over Fq[x]

Let p(x,y)=∑bj(x)yj∈Fq[x,y], R:=Fq[x,y]/(p(x,y)) and . For f∈R, the expansion f=∑di(x)yi (di(x)∈N) is called a y-adic expansion. Scheicher and Thuswaldner characterized those p(x,y) which render finite and periodic y-adic expansions. Here bounds on the lengths of elements whose expansions are finite, on the periods of elements with periodic expansions, conditions for a periodic expansion to have a prescribed period are determined and certain elements with infinite, non-periodic expansions are investigated.