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Determinantal numerical semigroup ring conjecture

For every numerical semigroup H=⟨a1,…,an⟩H=\langle a_1,\ldots,a_n\rangle minimally generated by nn elements, let IH=ker⁡(k[x1,…,xn]→k[t])I_H=\ker\bigl(k[x_1,\ldots,x_n]\to k[t]\bigr), where xi↦taix_i\mapsto t^{a_i}. Then IHI_H has a determinantal presentation if and only if there exist integers h≥0h\ge 0 and α>0\alpha>0 such that the set of pseudo-Frobenius numbers of HH is PF⁡(H)={h+α,h+2α,…,h+(n−1)α}\operatorname{PF}(H)=\{h+\alpha,h+2\alpha,\ldots,h+(n-1)\alpha\}, where PF⁡(H)={x∉H:x+ai∈H for every i}\operatorname{PF}(H)=\{x\notin H:x+a_i\in H\text{ for every }i\}.

References

Primary source

arXiv

Additional references

commutative-algebranumerical-semigroups
Posted · edited

Progress summary

Refreshed
Claimed progress

A 2026 paper advances the conjecture with a criterion and several special cases, but the general statement remains open.

The conjecture of Cuong, Kien, Matsuoka, and Truong proposes an equivalence between determinantal presentations of numerical semigroup rings and arithmetic progressions of pseudo-Frobenius numbers. The unrestricted converse is open in the retrieved evidence.

Known results

  • The implication from a determinantal presentation to an arithmetic progression was already known.
  • The converse is known in special families, including almost symmetric semigroups, maximal embedding dimension, generalized repunit semigroups, and stretched numerical semigroup rings.
  • One earlier paper proves the converse when a0/2+1≤na_0/2+1\leq n.
  • A later criterion reduces the general converse to factorizations of maximal Apéry-set elements.

September 2026 conditional advances

Submitted on September 16, 2026, Murai and Takahashi gave a numerical criterion and verified the conjecture in embedding dimension four under a unique-factorization hypothesis. Do Van Kien’s related work applies the added hypothesis to affine-orbit numerical semigroups, producing determinantal ideals and graded resolutions. These are advances, not a proof of the unrestricted conjecture.

Current status (as of October 2026): Several conditional and special cases are established, but the unrestricted conjecture remains open.

Sources

Solutions 0

No solutions have been posted yet.