弱牛顿非退化二项式理想

中文摘要 AI 辅助

多项式环 $R=k[x_1,\dots,x_n]$ 的理想 $I$ 若其整闭包 $\overline{I}$ 是单项式理想，则称为弱牛顿非退化（weakly Newton-nondegenerate，简称弱 NND）。我们研究代数闭域上二次二项式理想族 $I=(x_1^2+\epsilon_1x_{a_1}x_{b_1},\\ \dots,\\ x_n^2+\epsilon_nx_{a_n}x_{b_n})$（其中 $\epsilon_i\in\{\pm1\}$，$a_i\neq b_i$）的弱牛顿非退化性。我们证明：$I$ 是弱 NND 当且仅当 $\overline{I}=\mathfrak{m}^2$，当且仅当给定的生成元构成正则序列，当且仅当支撑模式与符号模式的组合条件成立：不存在非空子集 $S\subseteq\{1,\dots,n\}$ 同时对支撑数据封闭且对关联关系格平凡。最后一个等价性基于 Eisenbud 和 Sturmfels 风格下可除阿贝尔群上单项式方程组的可解性准则。作为应用，我们对 $n=3$ 的所有此类理想进行了分类。

英文摘要

An ideal $I$ of a polynomial ring $R=k[x_1,\dots,x_n]$ is called weakly Newton-nondegenerate, or weakly NND, if its integral closure $\overline{I}$ is a monomial ideal. We study weak Newton nondegeneracy for the family of quadratic binomial ideals \[ I=(x_1^2+ε_1x_{a_1}x_{b_1},\ \dots,\ x_n^2+ε_nx_{a_n}x_{b_n}),\qquad ε_i\in\{\pm1\},\ a_i\neq b_i, \] over an algebraically closed field. We prove that $I$ is weakly NND if and only if $\overline I=\mathfrak{m}^2$, if and only if the given generators form a regular sequence, and if and only if an explicit combinatorial condition on the pair (support pattern, sign pattern) holds: no nonempty subset $S\subseteq\{1,\dots,n\}$ is simultaneously closed for the support data and sign-trivial for the associated lattice of relations. The last equivalence rests on a solvability criterion for systems of monomial equations over a divisible abelian group, in the spirit of Eisenbud and Sturmfels. As an application, we classify all such ideals for $n=3$.

发表机构

• The University of Osaka（大阪大学）
• Tulane University（杜兰大学）

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