Let H=langle a_1,…,a_n rangle be a numerical semigroup minimally generated by a_1,…,a_n, let S=k[x_1,…,x_n], and let I_H=ker bigl(S→ k[t], x_i mapsto t^a_ibigr) be the defining ideal of the numerical semigroup ring k[H]. Then I_H admits a determinantal presentation I_H=I_2(M) for some 2× n matrix M over S if and only if the pseudo-Frobenius numbers of H, operatornamePF(H)=x∈Zsetminus H:x+h∈ H for every h∈ H setminus0, form an arithmetic progression; equivalently, there exist integers b and d>0 such that operatornamePF(H)=b,b+d,…,b+(r-1)d, where r=|operatornamePF(H)|.

Cuong–Kien–Matsuoka–Truong conjecture

Let $H=\langle a_1,\ldots,a_n\rangle$ be a numerical semigroup minimally generated by $a_1,\ldots,a_n$ , let $S=k[x_1,\ldots,x_n]$ , and let $I_H=\ker\bigl(S\to k[t],\ x_i\mapsto t^{a_i}\bigr)$ be the defining ideal of the numerical semigroup ring $k[H]$ . Then $I_H$ admits a determinantal presentation $I_H=I_2(M)$ for some $2\times n$ matrix $M$ over $S$ if and only if the pseudo-Frobenius numbers of $H$ , $\operatorname{PF}(H)=\{x\in\mathbb{Z}\setminus H:x+h\in H\text{ for every }h\in H\setminus\{0\}\}$ , form an arithmetic progression; equivalently, there exist integers $b$ and $d>0$ such that $\operatorname{PF}(H)=\{b,b+d,\ldots,b+(r-1)d\}$ , where $r=|\operatorname{PF}(H)|$ .

References

Primary source

arXiv

Additional references

A criterion for determinantal presentations of numerical semigroup rings — arXiv — Satoshi Murai, Kou Takahashi

commutative-algebra numerical-semigroups

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new paper proves the conjecture in important special cases, but the general question remains open.

The conjecture asserts an equivalence between a determinantal presentation of a numerical-semigroup ring and the pseudo-Frobenius numbers forming an arithmetic sequence. The general converse is not settled.

Known results

The forward implication was established before Takahashi's December 2025 paper.
Takahashi's paper proves the converse when $\frac{a_0}{2}+1\le n$ , where $a_0$ is the multiplicity and $n$ the embedding dimension.
Earlier verified families include almost symmetric semigroups, maximal-embedding-dimension semigroups, generalized repunit semigroups, and stretched numerical-semigroup rings.

September 16, 2026 structural reduction

A report on Satoshi Murai and Kou Takahashi's paper gives a reduction to factorizations of maximal Apéry-set elements and a proof in embedding dimension four under an additional uniqueness assumption. This is substantive progress, but it does not resolve the full equivalence.

Current status (as of September 2026): The forward implication and several special cases are established, while the converse remains open in general; the newly reported reductions and embedding-dimension-four result are unverified here.

Sources

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