# randomShellableIdeal -- Produces a ideal from a random shellable simplicial complex

## Synopsis

• Usage:
I = randomShellableIdeal(R,m,k)
• Inputs:
• R, a ring, a polynomial ring
• m, an integer, dimension of facets in shellable complex
• k, an integer, the degree of the shellable complex
• Outputs:
• I, , the Stanley-Reisner ideal of a random shellable complex

## Description

The Stanley-Reisner ideal of a shellable simplicial complex is always Cohen-Macaulay; the converse is not true, although, to paraphrase Arnol'd, square-free monomial ideals that have a serious reason to be Cohen-Macaulay generally do come from shellable complexes.

The program makes a (Cohen-Macaulay) square-free monomial ideal from the Stanley-Reisner ideal of a random shellable simplicial complex. simplicial complex relies on the code for producing random shellable simplicial complexes; see randomShelling for a description.

 i1 : R = ZZ/101[x_0..x_4]; i2 : I = randomShellableIdeal(R,2,6) o2 = monomialIdeal (x x , x x x , x x x x ) 0 2 0 3 4 1 2 3 4 o2 : MonomialIdeal of R

## Caveat

No claim is made on the distribution of the ideal.

## Ways to use randomShellableIdeal :

• "randomShellableIdeal(Ring,ZZ,ZZ)"

## For the programmer

The object randomShellableIdeal is .