几乎对称的几乎完全交数值半群的结构
The structure of almost symmetric almost complete intersection numerical semigroups
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中文总结 AI 辅助
本文证明几乎对称且几乎完全交的数值半群具有级联矩阵结构，并由此得到刚性定理、级联多项式及Herzog–Watanabe问题的肯定解答。
中文摘要 AI 辅助
我们证明了几乎对称且几乎完全交的数值半群H的一个结构定理。具体来说，我们证明了H的行分解（RF）矩阵必须具有一种高度规则的结构，我们称之为级联矩阵。因此，相关半群环k[H]的定义理想I_H也表现出高度规则的结构，该结构源于这个级联矩阵。此外，RF矩阵和I_H的二项式极小生成集都是唯一的。反之，我们证明该结构完全刻画了几乎对称的几乎完全交数值半群：对每个级联矩阵M，我们关联一个幺半群H，并且只要这是一个数值半群，我们就证明它是伪对称的、几乎完全交的，并以M作为其RF矩阵。作为我们研究的推论，我们获得了几个额外的重要结果。1）刚性定理：如果一个几乎完全交半群是几乎对称的，那么它必然具有奇嵌入维数并且是伪对称的。该结果可被视为Kunz定理之后的“下一步”，Kunz定理指出几乎完全交半群绝不可能是对称的。2）级联多项式：对每个正奇数e，我们构造一个具有整数系数的多元无平方多项式P_e，它由变量的级联矩阵产生。我们给出了其系数的枚举解释，从而证明了它们的非负性。3）Herzog–Watanabe问题：在证明主定理的过程中，我们证明了任意数值半群H的每个极小关系都可以通过从H的某个RF矩阵中减去两行而获得，从而肯定地回答了Herzog和Watanabe在2019年提出的一个问题。
英文摘要
We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. Specifically, 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 associated semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Moreover, both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, we show that 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 it is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several additional key results. 1) A rigidity theorem: 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 can be regarded as the ``next step'' after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) 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. 3) Herzog--Watanabe question: en route to proving the main theorem, 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.
发表机构
- Nippon Institute of Technology（日本工业大学）
- University of Graz（格拉茨大学）
- Meiji University（明治大学）
- Politecnico di Milano（米兰理工大学）
- Nihon University（日本大学）
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