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. 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.