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arXiv · 2412.05945

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

Liuqing Yang·Zexin Wang
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Abstract

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.

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Liuqing Yang, Zexin Wang. 2024-12-08. The Multiplicity of Powers of a Class of Non-Square-free Monomial Ideals. https://arxiv.org/abs/2412.05945

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