Is the number of subrings of index $p^e$ in $\mathbb{Z}^n$ polynomial in $p$?

Abstract

It is well-known that for each fixed n and e, the number of subgroups of index p e in Z n is a polynomial in p. Is this true for subrings in Z n of index p e ? Let f n (k) denote the number of subrings of index k in Z n . We can define the subring zeta function overIs this zeta function uniform? These two questions are closely related.In this paper, we describe what is known about these questions, and we make progress toward answering them in a couple ways. First, we describe the connection between counting subrings of index p e in Z n and counting the solutions to a corresponding set of equations modulo various powers of p. We then show that the number of solutions to certain subsets of these equations is a polynomial in p for any fixed n. On the other hand, we give an example for which the number of solutions to a certain subset of equations is not polynomial. Finally, we give an explicit polynomial formula for the number of 'irreducible' subrings of index p n+2 in Z n .