The saturation number of monomial ideals

TL;DR

This paper investigates the saturation number of monomial ideals, providing explicit formulas and computations for irreducible, ordinary, and symbolic powers, especially in two-variable cases, enhancing understanding of their algebraic properties.

Contribution

It introduces methods to compute the saturation number for irreducible monomial ideals and their powers, extending to ordinary and symbolic powers, with explicit formulas in two variables.

Findings

Computed saturation number for irreducible monomial ideals.

Derived formulas for powers and symbolic powers of monomial ideals.

Provided explicit saturation number formula for two-variable monomial ideals.

Abstract

Let $S=\mathbb{K}[x_1,\ldots, x_n]$ be the polynomial ring over a field $\mathbb{K}$ and $\mathfrak{m}= (x_1, \ldots, x_n)$ be the irredundant maximal ideal of $S$. For an ideal $I \subset S$, let $\mathrm{sat}(I)$ be the minimum number $k$ for which $I \colon \mathfrak{m}^k = I \colon \mathfrak{m}^{k+1}$. In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appearing in an irreducible decomposition of these ideals. Moreover, we give an explicit formula for the saturation number of monomial ideals in two variables.