---
title: 'DetAS-X: Ideal-Counting Zeta Function Analysis'
url: https://www.emergentmind.com/topics/detas-x
type: topic
---

# DetAS-X: Ideal-Counting Zeta Function Analysis

Searching arXiv for the specified paper to ground the article in the source record.
The arXiv record [1608.08508] studies the ideal-counting zeta function of the ring generated by an integral square matrix \(B\) when \(B\) has exactly three integral eigenvalues. The central reduction identifies the subring \((B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}\) with a quotient of \(\mathbb{Z}[x]\) by a cubic polynomial, and then normalizes the problem to the ring
\[
\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],
\]
where \(0,\alpha,\beta\in\mathbb{Z}\) are distinct. The main objective is to compute the Dirichlet series
\[
\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},
\]
with \(a_n\) equal to the number of ideals of index \(n\) in \((B)\), and to make the corresponding Euler factors explicit [1608.08508].

## 1. Algebraic formulation and normalization

For an integral square matrix \(B\), the subring \((B)\subseteq M_n(\mathbb{Z})\) generated by \(B\) is a free \(\mathbb{Z}\)-module of finite rank, so only finitely many ideals can occur at each finite index. This makes the ideal-counting Dirichlet series \(\zeta_{(B)}(s)\) well-defined [1608.08508].

The key algebraic input is that when the minimal polynomial of \(B\) over \(\mathbb{Q}\) is
\[
m_B(x)=x(x-\alpha)(x-\beta),
\]
with \(0,\alpha,\beta\) distinct integers, one has
\[
(B)\cong \mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x].
\]
A further normalization uses the isomorphism
\[
\mathbb{Z}[x]/m(x)\mathbb{Z}[x]\cong \mathbb{Z}[x]/m(x+y)\mathbb{Z}[x]
\]
for any \(y\in \mathbb{Z}\), reducing the analysis to the above form [1608.08508].

This reduction is structurally important because it replaces a matrix-theoretic counting problem by a ring-theoretic one. A plausible implication is that the spectral constraint of “exactly three integral eigenvalues” is valuable not only for graph-theoretic examples but also because it places the ambient ring in a tractable cubic quotient framework.

## 2. Global zeta factorization and the role of bad primes

A Solomon-type local-global decomposition is used. If \(A\) is a semisimple \(\mathbb{Q}\)-algebra, \(\Lambda\) a \(\mathbb{Z}\)-order in \(A\), and \(L\) a \(\Lambda\)-lattice, then there are local rational functions \(d_p(x)\) such that
\[
\zeta_\Lambda(s)=\prod_{p\in B} d_p(p^{-s})\,\zeta_V(s)_p,
\]
where \(B\) is the finite set of primes at which \(\Lambda\) is not maximal, and \(\zeta_V(s)\) is the ambient zeta factor coming from the semisimple algebra decomposition [1608.08508].

In the cubic quotient under consideration,
\[
R=\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],
\]
the bad primes are exactly the primes dividing
\[
\alpha\beta(\beta-\alpha).
\]
Moreover,
\[
\zeta_V(s)=\zeta(s)^3,\qquad \zeta_V(s)_p=(1-p^{-s})^{-3}.
\]
Hence
\[
\zeta_R(s)=\left(\prod_{p\mid \alpha\beta(\beta-\alpha)} d_p(p^{-s})\right)\zeta(s)^3
\]
[1608.08508].

This factorization isolates all nontrivial arithmetic in finitely many Euler factors. For primes \(p\nmid \alpha\beta(\beta-\alpha)\), the local factor is simply \((1-p^{-s})^{-3}\), so away from the bad set the ideal-counting behavior is the same as that of \(\mathbb{Z}^3\). This suggests that the singular local structure of the cubic quotient is concentrated exactly where the three roots \(0,\alpha,\beta\) fail to remain pairwise distinct modulo \(p\).

## 3. Local model over \(\mathbb{Z}_p\)

The local counting problem is formulated over
\[
R_p=\mathbb{Z}_p[x]/x(x-\alpha)(x-\beta)\mathbb{Z}_p[x].
\]
The authors fix the \(\mathbb{Z}_p\)-basis
\[
\{1,\ x,\ x(x-\alpha)\}
\]
of \(R_p\), and parametrize every full ideal \(I\subseteq R_p\) as a free \(\mathbb{Z}_p\)-module of rank \(3\) with basis
\[
\left\{p^{r_1}+a_1x+a_2x(x-\alpha),\; p^{r_2}x+a_3x(x-\alpha),\; p^{r_3}x(x-\alpha)\right\},
\]
for some \((r_1,r_2,r_3)\in\mathbb{N}^3\) and \(a_1,a_2,a_3\in\mathbb{Z}_p\) [1608.08508].

The triple \((r_1,r_2,r_3)\) is uniquely determined by the ideal and serves as its type. The index of such an ideal is
\[
[R_p:I]=p^{r_1+r_2+r_3}.
\]
The ideal condition \(xI\subseteq I\) becomes the congruence system
\[
p^{r_2}+a_3(\beta-\alpha)\in p^{r_3}\mathbb{Z}_p \tag{5}
\]
\[
p^{r_1}+\alpha a_1\in p^{r_2}\mathbb{Z}_p \tag{6}
\]
\[
(p^{r_2}-\alpha a_3)a_1+\beta p^{r_2}a_2-p^{r_1}a_3\in p^{r_2+r_3}\mathbb{Z}_p. \tag{7}
\]
The counting problem is then converted into a sum over admissible valuation data and residue parameters [1608.08508].

The local behavior depends on the \(p\)-adic valuations \([\alpha]\), \([\beta]\), and \([\beta-\alpha]\), where for \(a\in\mathbb{Z}_p\), \([a]\) denotes the nonnegative integer such that \(a\mathbb{Z}_p=p^{[a]}\mathbb{Z}_p\), and \([0]=\infty\). By symmetry, one may assume
\[
[\alpha]\le [\beta]
\]
in the local analysis [1608.08508].

## 4. Parametrization of admissible coefficients

Two auxiliary sets organize the local counting. First,
\[
S(r_2,r_3)
\]
denotes the set of \(a_3\in\mathbb{Z}_p\) satisfying (5). Second,
\[
S(r_1,r_2,r_3,a_3)
\]
denotes the set of pairs \((a_1,a_2)\) satisfying (6) and (7). The number of ideals of type \((r_1,r_2,r_3)\) is therefore
\[
\sum_{a_3\in S(r_2,r_3)} |S(r_1,r_2,r_3,a_3)|
\]
[1608.08508].

A decisive lemma gives an explicit description of \(S(r_2,r_3)\). Writing \([\beta-\alpha]=v\),
\[
S(r_2,r_3)= \begin{cases} \varnothing, & r_2<\min\{v,r_3\},\\[4pt] \mathbb{Z}_p, & r_3\le \min\{v,r_2\},\\[4pt] p^{r_3-v}\mathbb{Z}_p, & v<r_3<r_2,\\[4pt] p^{r_3-v}\mathbb{Z}_p-p^{r_2-v}\mathbb{Z}_p, & v<r_2<r_3. \end{cases}
\]
This is the first major case split in the argument [1608.08508].

For a fixed \(a\in S(r_2,r_3)\), the allowable values of \(r_1\) are described by
\[
C(r_2,r_3,a)=\{r\in\mathbb{N}\mid r\ge \min\{[p^{r_2}-\alpha a],\,2[\alpha],\, [\alpha]+r_3\}\},
\]
and the number of solutions in \((a_1,a_2)\) is
\[
|S(r_1,r_2,r_3,a)|=p^{\min\{[p^{r_2}-\alpha a],\,2[\alpha],\, [\alpha]+r_3\}}.
\]
Once \(a_3\) is fixed, the count of admissible \((a_1,a_2)\) is therefore a pure power of \(p\) [1608.08508].

Further lemmas refine this by splitting \(S(r_2,r_3)\) according to the valuation of \(a_3\), computing \([p^{r_2}-\alpha a]\) in the relevant regimes, and decomposing into sets \(X_j\) where \([p^{r_2}-\alpha a]=r_2+j\) with exact cardinalities. This is what permits explicit Euler factors rather than only an algorithmic procedure [1608.08508].

## 5. Regionwise summation and local generating functions

The local ideal-counting function is defined by
\[
E(r_2,r_3)=\sum_{a_3\in S(r_2,r_3)} \sum_{r_1\in C(r_2,r_3,a_3)} |S(r_1,r_2,r_3,a_3)|\,p^{-s(r_1+r_2+r_3)}.
\]
With the notation
\[
x=p^{-s},
\]
this becomes a rational-function computation in \(x\) [1608.08508].

Theorem 3.7 partitions the \((r_2,r_3)\)-plane into six regions
\[
R_1,\dots,R_6
\]
according to the relative sizes of \(r_2,r_3,[\alpha],[\beta-\alpha]\), and \(E(r_2,r_3)\) has a uniform closed form on each region [1608.08508].

Among the explicit cases proved are the following:

- If \(r_2\ge [\beta-\alpha]+[\alpha]\), then
  \[
  E(r_2,r_3)=p^{2[\alpha]}\sum_{r_1\ge 2[\alpha]} p^{-s(r_1+r_2+r_3)}.
  \]

- If \([\beta-\alpha]<r_3\) and \([\beta-\alpha]<r_2<\min\{[\beta-\alpha]+[\alpha],r_3\}\), then
  \[
  E(r_2,r_3)=p^{r_2-[\beta-\alpha]+[\alpha]}\sum_{r_1\ge r_2-[\beta-\alpha]+[\alpha]}p^{-s(r_1+r_2+r_3)}.
  \]

The delicate finite region \(R_6\) requires the finer decomposition from Lemma 2.11 [1608.08508].

This regionwise organization is methodologically significant because it converts a valuation-sensitive ideal classification into a finite family of geometric-series evaluations. A plausible implication is that the principal technical obstacle is not the final summation but the precise stratification of \(p\)-adic parameter space.

## 6. Explicit formulas and graph-theoretic instances

The global zeta function always takes the form
\[
\zeta_R(s)=\left(\prod_{p\mid \alpha\beta(\beta-\alpha)} d_p(x)\right)\zeta(s)^3,
\qquad x=p^{-s},
\]
with \(d_p(x)\in\mathbb{Z}[x]\) determined explicitly by the valuations \([\alpha]\), \([\beta]\), and \([\beta-\alpha]\) [1608.08508].

Several valuation patterns are treated in closed form.

| Valuation pattern | Explicit formula |
|---|---|
| \(([\alpha],[\beta],[\beta-\alpha])=(0,0,k)\) | \(\displaystyle \zeta_R(s)=\frac{1-p^k x^{2k}}{1-px^2}(1-x)^{-2}+p^k x^{2k}(1-x)^{-3}\) |
| \((1,1,1)\) | \(\displaystyle \zeta_R(s)=\frac{p^3x^6-2p^2x^5+(p^2+p)x^4+(p^2-2p)x^3+(p+1)x^2-2x+1}{(1-x)^3}\) |
| \((1,1,2)\) | \(\displaystyle \zeta_R(s)=\frac{p^4x^8-2p^3x^7+(p^3+p^2)x^6+(p^3-p^2)x^5+px^4+(p^2-2p)x^3+(p+1)x^2-2x+1}{(1-x)^3}\) |

For the pattern \((0,0,k)\), the paper states the formula in Proposition 4.1 in equivalent simplified forms. The listed expression follows the closed form given there [1608.08508].

A prominent example is the Petersen graph. Its adjacency matrix has eigenvalues \(3,1,-2\), and after shifting the corresponding ring is
\[
\mathbb{Z}[x]/x(x+3)(x-2)\mathbb{Z}[x].
\]
Here the relevant valuations satisfy
\[
([\alpha],[\beta],[\beta-\alpha])=(0,0,1),
\]
so Proposition 4.1 gives
\[
\zeta_{(B)}(s)=\prod_{p=2,3,5}\left(1-p^{-s}+p^{1-2s}\right)\zeta(s)^3
\]
[1608.08508].

The paper also states that, for the adjacency matrix of the cycle of length \(4\), a formula is obtained from Proposition 4.3 and matches a computer-assisted earlier computation. It further remarks that the same framework applies to many graph adjacency matrices with exactly three integral eigenvalues, including
\[
K_{3,3},\ K_{2,2,2},\ K_{5,5},\ \mathrm{Paley}(9),\ T(5),\ K_{6,6},\ K_{4,4,4},\ K_{3,3,3,3},\ K_{2,2,2,2,2,2},\ \mathrm{Sp}(4,2),\ T(6),
\]
provided the local valuation pattern matches one of the treated cases [1608.08508].

## 7. Multiplicativity and mathematical significance

Because the zeta function is an Euler product, the coefficients \(a_n\) are multiplicative. Equivalently,
\[
\zeta_R(s)=\prod_p \zeta_{R,p}(s),
\]
with each \(\zeta_{R,p}(s)\) a rational function in \(p^{-s}\), so the coefficients are determined by the counts at prime powers together with multiplicativity across coprime \(n\) [1608.08508].

For primes \(p\nmid \alpha\beta(\beta-\alpha)\), the local factor is
\[
(1-p^{-s})^{-3},
\]
and the nontrivial arithmetic is localized at finitely many primes. This sharply separates generic behavior from singular local corrections [1608.08508].

The overall derivation proceeds by algebraic reduction to the cubic quotient, localization at \(p\), parametrization of full ideals by a triangular \(\mathbb{Z}_p\)-basis, translation of closure under multiplication by \(x\) into congruences (5)–(7), valuation-theoretic analysis of the solution sets, partition of the \((r_2,r_3)\)-plane into six regions, summation of the resulting geometric series, and assembly of the Euler product [1608.08508]. In methodological terms, the work connects spectral data of integral matrices—particularly adjacency matrices with three integral eigenvalues—to explicit ideal zeta functions of associated \(\mathbb{Z}\)-orders. This suggests a concrete bridge between algebraic graph theory, the arithmetic of orders, and local counting methods over \(\mathbb{Z}_p\).

Source: https://www.emergentmind.com/topics/detas-x