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PreprintCC BY 4.0arXiv

Admissible bases and generic lattice ideals

  • Anargyros Katsabekis
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Abstract

Let L⊆ℤ4 be a rank-three lattice possessing an admissible basis, and let S=K[x1,…,x4] be the polynomial ring over a field K. We show that the associated lattice ideal IL ⊂ S is generic and that S/IL is a Cohen--Macaulay ring. We also identify the neighbors of the origin in L and the indispensable binomials of IL. Moreover, we prove that the Betti numbers of S/IL are β0(S/IL)=1, β1(S/IL)=7, β2(S/IL)=12, and β3(S/IL)=6. In addition, we show that IL is a set-theoretic complete intersection over an arbitrary field.

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Cite

BibTeXRIS
BibTeX
@misc{katsabekis2026admissible,
  title = {{Admissible bases and generic lattice ideals}},
  author = {Anargyros Katsabekis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.18697},
  eprint = {2607.18697},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.18697},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Admissible bases and generic lattice ideals
AU  - Anargyros Katsabekis
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2607.18697
UR  - https://arxiv.org/abs/2607.18697
AB  - Let L⊆ℤ^4 be a rank-three lattice possessing an admissible basis, and let S=K[x_1,…,x_4] be the polynomial ring over a field K. We show that the associated lattice ideal I_L ⊂ S is generic and that S/I_L is a Cohen--Macaulay ring. We also identify the neighbors of the origin in L and the indispensable binomials of I_L. Moreover, we prove that the Betti numbers of S/I_L are β_0(S/I_L)=1, β_1(S/I_L)=7, β_2(S/I_L)=12, and β_3(S/I_L)=6. In addition, we show that I_L is a set-theoretic complete intersection over an arbitrary field.
ER  -