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

TL;DR

This paper derives formulas for the multiplicity of powers of certain monomial ideals, including special powers and edge ideals of weighted oriented graphs, generalizing previous results in algebraic geometry.

Contribution

It introduces new formulas for the multiplicity of powers of monomial ideals with irreducible primary components, extending to special powers and edge ideals of weighted oriented graphs.

Findings

Derived a formula for multiplicity of powers of monomial ideals with irreducible primary components.

Provided explicit formulas for special powers of monomial ideals.

Extended results to edge ideals of weighted oriented graphs.

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.