Unit-scaled determinant-count conjecture for circulant matrices over finite chain rings

Let $R$ be a commutative finite chain ring (CFCR) of nilpotency index $e$ , let $n$ be a positive integer, and suppose that the maximal ideal of $R$ is generated by $\gamma$ . Write $U(R)$ for the group of units of $R$ , and let $c_n(R,a)$ denote the number of $n\times n$ circulant matrices over $R$ with determinant $a$ . Unit-scaling conjecture. For every unit $b\in U(R)$ and every integer $s$ with $0\leq s\leq e$ ,

c_n(R,\gamma^s)=c_n(R,b\gamma^s).

This number may be zero. The conjecture concerns invariance of determinant counts under multiplication by units; the source gives no resolution or supporting result beyond presenting it as a conjecture.

Progress summary

Solved

A reader-posted calculation claims the conjecture is false in two dimensions, while the published work proves it only when the matrix size avoids the residue-field characteristic.

Jitman posed the unit-scaling conjecture in 2020: for a commutative finite chain ring, determinant counts at $\gamma^s$ and at $b\gamma^s$ should agree for every unit $b$ and every admissible $s$ .

Known results

Jitman (2020) proves $c_n(R,b\gamma^s)=c_n(R,\gamma^s)>0$ when $\gcd(n,q)=1$ .
The same paper leaves the cases $\gcd(n,q)\ne 1$ open and notes that the common count may be zero.
For certain singular cases with $n\mid(q-1)$ , Jitman gives explicit determinant-count formulas.

Posted attempt

A reader claims a complete counterexample over $R=\mathbb{F}_2[\varepsilon]/(\varepsilon^2)$ with $n=2$ : it computes $c_2(R,1)=8$ but $c_2(R,1+\varepsilon)=0$ . The calculation has not been independently verified.

Current status (as of August 2026): The conjecture is proved for $\gcd(n,q)=1$ , while its unrestricted form has an unverified claimed counterexample and no confirmed resolution.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Somphong Jitman, “Determinants of some Special Matrices over Commutative Finite Chain Rings”, arXiv:2007.14123 (2020).

1 week ago

Take the commutative finite chain ring

R=\mathbb F_2[\varepsilon]/(\varepsilon^2).

Its ideals are $(0)\subset(\varepsilon)\subset R$ , its nilpotency index is $e=2$ , its residue field has order $q=2$ , and its units are $1$ and $1+\varepsilon$ .

For $n=2$ , every circulant matrix over $R$ is

A=\begin{pmatrix}a&b\b&a\end{pmatrix}.

Since the characteristic is two,

\det A=a^2-b^2=(a+b)^2.

If $a+b=c+d\varepsilon$ with $c,d\in\mathbb F_2$ , then

\det A=(c+d\varepsilon)^2=c.

Thus every circulant determinant belongs to $\{0,1\}$ , and none equals the unit $1+\varepsilon$ .

Choose the maximal-ideal generator $\gamma=\varepsilon$ , the admissible exponent $s=0$ , and the unit $u=1+\varepsilon$ . Each prescribed value of $a+b$ arises from exactly four ordered pairs $(a,b)$ . Exactly two possible sums, $1$ and $1+\varepsilon$ , have constant coefficient $1$ . Therefore

c_2(R,\gamma^0)=c_2(R,1)=8, \qquad c_2(R,u\gamma^0)=c_2(R,1+\varepsilon)=0.

Hence

c_2(R,\gamma^s)\ne c_2(R,u\gamma^s),

contradicting the conjectured unit-scaling equality.

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Shivam Patel · 2 days ago