Is the number of subrings of index pe in ℤ n polynomial in p?

Abstract

For fixed n and e, the number of subgroups of index in is polynomial in p. Is this true for subrings in of index ? Equivalently, is the subring zeta function uniform? In this paper, we make progress toward answering these questions. First, we connect counting subrings of index in to counting solutions to equations modulo various powers of p. We show some subsets of these equations have a polynomial number of solutions, but others do not. Finally, we give an explicit polynomial formula for the number of ‘irreducible’ subrings of index in .