components(ChordalNet,ZZ) -- components of a chordal network

Synopsis

Function: components
Usage:

components(N)

components(N,k)
Inputs:
- N, an instance of the type ChordalNet,
- k, an integer, (optional)
Outputs:
- a list, of components of the network

Description

Enumerates the (maximal) components of a chordal network. If the optional argument $k$ is given, then only the components in the top $k$ dimensions are computed.

i1 : I = toLex edgeIdeal cycleGraph 8

o1 = ideal (x x , x x , x x , x x , x x , x x , x x , x x )
             1 2   2 3   3 4   4 5   5 6   6 7   1 8   7 8

o1 : Ideal of QQ[x ..x ]
                  1   8

i2 : N = chordalNet I;

i3 : chordalTria N;

i4 : codimCount N

      7     6      5     4
o4 = t  + 8t  + 13t  + 2t

o4 : ZZ[t]

i5 : components(N,1)

o5 = HashTable{4 => {ideal (x , x , x , x ), ideal (x , x , x , x )}}
                             1   3   5   7           2   4   6   8

o5 : HashTable

i6 : components(N)

o6 = HashTable{4 => {ideal (x , x , x , x ), ideal (x , x , x , x )}                                                                                                                                                                                }
                             1   3   5   7           2   4   6   8
               5 => {ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x )}
                             1   2   4   5   7           1   2   4   6   7           1   3   4   6   7           2   3   5   7   8           2   4   5   7   8           1   3   5   6   8           2   3   5   6   8           1   3   4   6   8
               6 => {}
               7 => {}

o6 : HashTable

Caveat

It is assumed that the chains of the network define prime ideals.

Ways to use this method: