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

TL;DR

This paper investigates whether the number of subrings of a given index in ^n is polynomial in p, exploring connections to solution counts of modular equations and providing explicit formulas for certain cases.

Contribution

It establishes links between subring counts and solutions to modular equations, and provides explicit polynomial formulas for specific subring counts.

Findings

Number of solutions to certain modular equations is polynomial in p for fixed n.

Counterexample shows some solution counts are not polynomial.

Explicit polynomial formula derived for irreducible subrings of index p^{n+2}.

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 $\mathbb{Z}^n$.