Unit determinant conjecture for circulant matrices over finite chain rings

About 6 years old · traced to 2020

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.

References

Primary source

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-supplied example claims the conjecture is false, but this counterexample has not been independently verified; the only published result covers the coprime case.

Jitman’s 2020 paper conjectures that every unit is the determinant of an $n\times n$ circulant matrix over every commutative finite chain ring, without assuming $\gcd(n,q)=1$ .

Known results

If $\gcd(n,q)=1$ , every element of $R$ , in particular every unit, occurs as a determinant (Jitman, 2020).
Without this coprimality condition, surjectivity can fail for nonunits: $2$ is not the determinant of any $2\times2$ matrix over $\mathbb{Z}_4$ (Jitman, 2020).

Posted attempt

A reader claims a counterexample with $R=\mathbb{F}_2[\varepsilon]/(\varepsilon^2)$ and $n=2$ : every determinant is in $\{0,1\}$ , so the unit $1+\varepsilon$ is omitted. The calculation is not independently verified.

Current status (as of August 2026): The coprime case is proved, while the general unit conjecture has an unverified claimed counterexample and therefore remains mathematically unsettled.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

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.