Skip to content
DraconiticResearch
Search Sources API
197,761 studies31 sourcesUpdated 58 min ago
PreprintCC BY 4.0arXiv

Arithmetic pseudo-Frobenius numbers and determinantal numerical semigroup rings

  • Do Van Kien
Open at sourcePDFDOICite

Abstract

Let H=⟨ a1,…,an⟩ be a numerical semigroup. A conjecture on numerical semigroup rings predicts that the defining ideal IH has a determinantal presentation if and only if the set of pseudo-Frobenius numbers of H has form {h+α,h+2α,…,h+(n−1)α} for h≥ 0, α>0. We prove this conjecture under the condition that h has a unique factorization in H. We also give characterization this unique-factorization condition. As an application, we consider the family of affine-orbit numerical semigroups determined by m=cAn−1+1, 1≤ c≤ a. We prove that the relevant element h has a unique factorization and the defining ideal is generated by the 2-minors of a 2× n-matrix. In particular, the Eagon--Northcott complex of the matrix mentioned above gives the graded minimal free resolution of k[H].

Related

  1. A criterion for determinantal presentations of numerical semigroup rings
    Satoshi Murai, Kou Takahashi
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. The Betti numbers and Golodness of numerical semigroup rings of Sally type
    Manav Batavia, Omkar Javadekar, Manohar Kumar
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Reflexive ideals of a numerical semigroup
    P. Campochiaro, M. D'Anna, P. A. Garcia-Sanchez
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Betti Sequences of Four-Generated Non-Almost-Symmetric Nearly Gorenstein Numerical Semigroup Rings
    Zhi-Lin Zhang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Frobenius Numbers Associated with Primitive Pythagorean Quadruples
    WonTae Hwang, Kyunghwan Song
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{kien2026arithmetic,
  title = {{Arithmetic pseudo-Frobenius numbers and determinantal numerical semigroup rings}},
  author = {Do Van Kien},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39218},
  eprint = {2609.39218},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39218},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Arithmetic pseudo-Frobenius numbers and determinantal numerical semigroup rings
AU  - Do Van Kien
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39218
UR  - https://arxiv.org/abs/2609.39218
AB  - Let H=⟨ a_1,…,a_n⟩ be a numerical semigroup. A conjecture on numerical semigroup rings predicts that the defining ideal I_H has a determinantal presentation if and only if the set of pseudo-Frobenius numbers of H has form {h+α,h+2α,…,h+(n−1)α} for h≥ 0, α>0. We prove this conjecture under the condition that h has a unique factorization in H. We also give characterization this unique-factorization condition. As an application, we consider the family of affine-orbit numerical semigroups determined by m=cA_n−1+1, 1≤ c≤ a. We prove that the relevant element h has a unique factorization and the defining ideal is generated by the 2-minors of a 2× n-matrix. In particular, the Eagon--Northcott complex of the matrix mentioned above gives the graded minimal free resolution of k[H].
ER  -