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 ${\mathbb{Z}}^n$ is a polynomial in $p$. Is this true for \emph{subrings} in ${\mathbb{Z}}^n$ of index $p^e$? Let $f_n(k)$ denote the number of subrings of index $k$ in ${\mathbb{Z}}^n$. We can define the subring zeta function over ${\mathbb{Z}}^n$ to be ${\zeta }_{{\mathbb{Z}}^n}^R(s)={\sum }_{k\ge 1}{f}_n(k)k^{-s}$. Is 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 ${\mathbb{Z}}^n$ 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 ${\mathbb{Z}}^n$.

34 pages including appendix