标题： 幂的重数的非平方自由单项式理想类 显示英文标题

标题： The Multiplicity of Powers of a Class of Non-Square-free Monomial Ideals

摘要： 设$R = \mathbb{K}[x_1, \ldots, x_n]$为域$\mathbb{K}$上的多项式环，令$I \subseteq R$为高度为$h$的单项式理想。当$I$的不可约简的素分解中所有高度为$h$的素理想都是不可约的时，我们给出了$I$幂的重数的公式。 这是一个\cite[Theorem 1.1]{TV}的推广。 此外，我们给出了满足上述条件的单项式理想特殊幂的乘法性的公式。 在这里，对于一个整数$m>0$，单项式理想的$m$-次特殊幂指的是由其所有极小生成元的$m$-次幂生成的理想。 最后，我们明确提供了加权有向图的边理想的特殊幂的乘法性的公式。 显示英文摘要

摘要： Let $R = \mathbb{K}[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb{K}$, and let $I \subseteq R$ be a monomial ideal of height $h$. We provide a formula for the multiplicity of the powers of $I$ when all the primary ideals of height $h$ in the irredundant reduced primary decomposition of $I$ are irreducible. This is a generalization of \cite[Theorem 1.1]{TV}. Furthermore, we present a formula for the multiplicity of powers of special powers of monomial ideals that satisfy the aforementioned conditions. Here, for an integer $m>0$, the $m$-th special power of a monomial ideal refers to the ideal generated by the $m$-th powers of all its minimal generators. Finally, we explicitly provide a formula for the multiplicity of powers of special powers of edge ideals of weighted oriented graphs.