论文标题
$ \ Mathbb {z}^n $ polyenmial in $ p $中的索引$ p^e $的子语数是否？
Is the number of subrings of index $p^e$ in $\mathbb{Z}^n$ polynomial in $p$?
论文作者
论文摘要
众所周知，对于每个固定的$ n $和$ e $，$ \ mathbb {z}^n $中的索引$ p^e $的子组的数量是$ p $中的多项式。对于\ \ m athbb {z}^n $ of Index $ p^e $的\ emph {subrings}这是正确的吗？令$ f_n（k）$表示索引$ k $ in $ \ mathbb {z}^n $的子环数。我们可以通过$ \ mathbb {z}^n $定义Zeta函数为$ζ_ {\ Mathbb {Z}^n}^r（s）= \ sum_ {k \ ge 1} f_n（k）f_n（k）k^{ - s} $。这个Zeta功能均匀吗？这两个问题密切相关。 在本文中，我们描述了对这些问题的了解，并在以几种方式回答它们方面取得了进展。首先，我们描述了$ \ mathbb {z}^n $中计数索引$ p^e $的连接与将解决方案计算到相应的方程组模型的各种功率$ p $。然后，我们证明，对于任何固定的$ n $，这些方程的某些子集的解决方案数量是$ p $的多项式。另一方面，我们举了一个例子，该方程组的某个子集的解决方案的数量不是多项式。最后，我们为$ \ Mathbb {z}^n $中的索引$ p^{n+2} $的“不可约”子环给出了明确的多项式公式。
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 $ζ_{\mathbb{Z}^n}^R(s) = \sum_{k \ge 1} f_n(k)k^{-s}$. 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 $\mathbb{Z}^n$.
