Unit determinant conjecture for circulant matrices over finite chain rings

From papers

Let $R$ be a commutative finite chain ring (CFCR) of nilpotency index $e$ , and let $n$ be a positive integer. Let $C_n(R)$ denote the ring of $n\times n$ circulant matrices over $R$ , let $U(R)$ be the group of units of $R$ , and let $\det(A)$ denote the determinant of $A$ . Unit determinant conjecture. Every unit of $R$ occurs as the determinant of a circulant matrix over $R$ :

U(R)\subseteq\{\det(A)\mid A\in C_n(R)\}.

Equivalently, $c_n(R,a)>0$ for every unit $a\in U(R)$ . This extends the determinant-surjectivity result known under the restriction $\gcd(n,q)=1$ ; the source explains that surjectivity can fail for nonunits when this restriction is removed, but leaves the unit case open.

Progress summary

Open

The conjecture remains open: it is proved when the matrix size and residue-field characteristic are coprime, but no public proof or counterexample for the general unit case was found.

The 2020 paper formulates the assertion as Conjecture $5.3$ : every unit of a commutative finite chain ring should be the determinant of a circulant matrix of every positive size. It also records the unrestricted counting problem as Problem $5.4$ .

Known results

Every unit occurs when $\gcd(n,q)=1$ (Determinants of some Special Matrices over Commutative Finite Chain Rings, 2020).
Without $\gcd(n,q)=1$ , surjectivity can fail for nonunits; the paper gives the zero-divisor $2$ over $\mathbb{Z}_4$ as an example, but reports no unit counterexample.

Current status (as of August 2026): The coprime case is settled, while the unit-determinant conjecture for arbitrary $n$ and residue-field size remains open, with no verified later progress found.

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 $R=\mathbb F_2[\varepsilon]/(\varepsilon^2)$ and $n=2$ . This is a commutative finite chain ring: its ideals are

(0)\subset(\varepsilon)\subset R,

its nilpotency index is $e=2$ , and its residue field is $\mathbb F_2$ . Its units are $1$ and $1+\varepsilon$ .

Every $2\times2$ circulant matrix over $R$ has the form

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

Because $R$ has characteristic two,

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

Writing $a+b=c+d\varepsilon$ , with $c,d\in\mathbb F_2$ , gives

(a+b)^2=c^2+2cd\varepsilon+d^2\varepsilon^2=c\in\{0,1\}.

Consequently no circulant matrix has determinant $1+\varepsilon$ , although $1+\varepsilon\in U(R)$ . Hence

U(R)\not\subseteq\{\det A:A\in C_2(R)\},

disproving the conjecture.

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Shivam Patel · 1 day ago