# The saturation number of powers of graded ideals

## TL;DR

This paper introduces the concept of saturation number for graded ideals in polynomial rings, analyzing its properties and behavior for various classes of ideals, including monomial, principal Borel, and squarefree Veronese ideals.

## Contribution

It defines the saturation number for graded ideals and investigates its bounds and exact values for specific ideal classes, revealing linear and quasi-linear behaviors.

## Key findings

- The saturation number is linearly bounded for all ideals.
- For monomial ideals, the saturation number function is quasi-linear for large k.
- Explicit formulas are provided for principal Borel and squarefree Veronese ideals.

## Abstract

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ with maximal ideal $\frak{m}=(x_1,...,x_n)$, and let $I$ be a graded ideal of $S$. In this paper, we define the saturation number $\sat(I)$ of $I$ to be the smallest non-negative integer $k$ such that $I:\mm^{k+1}= I:\mm^k$. We show that $f(k)$ is linearly bounded, and that $f(k)$ is a quasi-linear function for $k\gg 0$, if $I$ is a monomial ideal. Furthermore, we show that $\sat(I^k)=k$ if $I$ is a principal Borel ideal and prove that $\sat(I_{d,n}^k) =\max\{l\:\; (kd-l)/(k-l) \leq n\},$ where $I_{d,n}$ is the squarefree Veronese ideal generated in degree $d$. \end{abstract}