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, a monomial ideal, 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 a method function.