含有给定指标的$x(x-α)(x-β)$的$\mathbb{Z}[x]$的理想数
The number of ideals of $\mathbb{Z}[x]$ containing $x(x-α)(x-β)$ with given index
摘要：
众所周知，一个连通正则图是强正则的当且仅当它的邻接矩阵恰好有三个特征值。令$B$表示整数方阵，$\langle B \rangle$表示由$B$生成的全矩阵环的子环。则$\langle B \rangle$是有限秩的自由$\mathbb{Z}$ -模，它保证了$\langle B \rangle$在给定的有限指标下只存在1/2个理想.因此，形式狄利克雷级数$ζ_{\langle B \rangle}(s)=\sum_{n\geq 1}a_n n^{-s}$是定义良好的，其中$a_n$是$\langle B \rangle$的具有指标$n$的理想数。在这篇文章中，我们的目标是找到一个明确的形式，$ζ_{\langle B \rangle}(s)$时，$B$正好有三个特征值，所有这些都是整数，例如，强正则图的邻接矩阵，它不是顶点数非平方的会议图。根据环的同构定理，$\langle B \rangle$同构于$\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$，其中$m(x)$是$B$在$\mathbb{Q}$上的最小多项式，对于每个$γ\in \mathbb{Z}$，$\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$同构于$\mathbb{Z}[x]/m(x+γ)\mathbb{Z}[x]$.因此，问题归结为计算具有给定有限指数的$\mathbb{Z}[x]/x(x-α)(x-β)\mathbb{Z}[x]$的理想数，其中$0,α$和$β$是不同的整数。
英文摘要：
It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $\langle B \rangle$ denote the subring of the full matrix ring generated by $B$. Then $\langle B \rangle$ is a free $\mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $\langle B \rangle$ with given finite index. Thus, the formal Dirichlet series $ζ_{\langle B \rangle}(s)=\sum_{n\geq 1}a_n n^{-s}$ is well-defined where $a_n$ is the number of ideals of $\langle B \rangle$ with index $n$. In this article we aim to find an explicit form of $ζ_{\langle B \rangle}(s)$ when $B$ has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, $\langle B \rangle$ is isomorphic to $\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$ where $m(x)$ is the minimal polynomial of $B$ over $\mathbb{Q}$, and $\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$ is isomorphic to $\mathbb{Z}[x]/m(x+γ)\mathbb{Z}[x]$ for each $γ\in \mathbb{Z}$. Thus, the problem is reduced to counting the number of ideals of $\mathbb{Z}[x]/x(x-α)(x-β)\mathbb{Z}[x]$ with given finite index where $0,α$ and $β$ are distinct integers.
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