The structure of almost symmetric almost complete intersection numerical semigroups
Abstract
We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. We show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that H is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several key results. 1) Rigidty: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result is a ``next step'' after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Huneke-Ulrich structure: the toric ideal can be realized as the ideal of entries of an alternating matrix multiplied by a vector of variables, and this yields the licci property. 3) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 4) Herzog--Watanabe question: we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.
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BibTeX
@misc{eto2026structure,
title = {{The structure of almost symmetric almost complete intersection numerical semigroups}},
author = {Kazufumi Eto and Naoyuki Matsuoka and Alessio Moscariello and Takahiro Numata and Alessio Sammartano and Kei-ichi Watanabe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.24880},
eprint = {2609.24880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24880},
}
RIS
TY - UNPB
TI - The structure of almost symmetric almost complete intersection numerical semigroups
AU - Kazufumi Eto
AU - Naoyuki Matsuoka
AU - Alessio Moscariello
AU - Takahiro Numata
AU - Alessio Sammartano
AU - Kei-ichi Watanabe
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.24880
UR - https://arxiv.org/abs/2609.24880
AB - We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. We show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that H is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several key results. 1) Rigidty: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result is a ``next step'' after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Huneke-Ulrich structure: the toric ideal can be realized as the ideal of entries of an alternating matrix multiplied by a vector of variables, and this yields the licci property. 3) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 4) Herzog--Watanabe question: we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.
ER -