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# stablePrimes -- compute the set of stable primes of a monomial ideal

## Synopsis

• Usage:
stablePrimes I
• Inputs:
• I, an ideal, a monomial ideal of polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
• Outputs:
• l, a list, all stable primes of $I$

## Description

By a classical result of Brodmann, $\text{Ass}(I^k)=\text{Ass}(I^{k+1})$ for all $k\gg0$. We denote the common set $\text{Ass}(I^k)$ for $k\gg0$ by $\text{Ass}^\infty(I)$. A prime ideal $\mathfrak{p}\subset S$ such that $\mathfrak{p}\in \text{Ass}(I^k)$ for all $k \gg 0$ is called a stable prime of $I$. This method computes $\text{Ass}^\infty(I)$, the set of the stable primes of $I$.

 i1 : S = QQ[x_1..x_3]; i2 : I = ideal(x_1*x_2,x_1*x_3,x_2*x_3); o2 : Ideal of S i3 : stablePrimes I o3 = {ideal (x , x ), ideal (x , x ), ideal (x , x ), ideal (x , x , x )} 2 1 3 1 3 2 1 2 3 o3 : List
 i4 : S = QQ[a..e]; i5 : I = ideal(a*d,a*e,b*d,c*d,c*e); o5 : Ideal of S i6 : stablePrimes I o6 = {ideal (c, b, a), ideal (d, c, a), ideal (e, d)} o6 : List

• isStablePrime -- test whether a prime ideal is a stable prime of a ideal $I$
• stableMax -- compute the set of stable primes of a monomial ideal that are maximal with respect to the inclusion.

## Ways to use stablePrimes :

• stablePrimes(Ideal)

## For the programmer

The object stablePrimes is .