-- -*- M2-comint -*- hash: 2646514380970138188

i1 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}})

o1 = MixedGraph{Bigraph => Bigraph{a => {d}}   }
                                   d => {a}
                Digraph => Digraph{b => {c, d}}
                                   c => {d}
                                   d => {}
                Graph => Graph{}

o1 : MixedGraph

i2 : R = gaussianRing G

o2 = R

o2 : PolynomialRing

i3 : compactMatrixForm =false;

i4 : S = covarianceMatrix(R)

o4 = |  s     s     s     s     |
     |   a,a   a,b   a,c   a,d  |
     |                          |
     |  s     s     s     s     |
     |   a,b   b,b   b,c   b,d  |
     |                          |
     |  s     s     s     s     |
     |   a,c   b,c   c,c   c,d  |
     |                          |
     |  s     s     s     s     |
     |   a,d   b,d   c,d   d,d  |

             4      4
o4 : Matrix R  <-- R

i5 : L = directedEdgesMatrix(R)

o5 = |  0  0    0     0   |
     |                    |
     |  0  0  l     l     |
     |         b,c   b,d  |
     |                    |
     |  0  0    0   l     |
     |               c,d  |
     |                    |
     |  0  0    0     0   |

             4      4
o5 : Matrix R  <-- R

i6 : W = bidirectedEdgesMatrix(R)

o6 = |  p       0     0   p     |
     |   a,a               a,d  |
     |                          |
     |    0   p       0     0   |
     |         b,b              |
     |                          |
     |    0     0   p       0   |
     |               c,c        |
     |                          |
     |  p       0     0   p     |
     |   a,d               d,d  |

             4      4
o6 : Matrix R  <-- R

i7 : M = gaussianParametrization(R)

o7 = |  p                0                                0                  
     |   a,a                                                                 
     |                                                                       
     |    0             p                             l   p                  
     |                   b,b                           b,c b,b               
     |                                                                       
     |                                              2                        
     |    0           l   p                        l   p    + p              
     |                 b,c b,b                      b,c b,b    c,c           
     |                                                                       
     |                                  2                                    
     |  p     l   l   p    + l   p     l   l   p    + l   l   p    + l   p   
     |   a,d   b,c c,d b,b    b,d b,b   b,c c,d b,b    b,c b,d b,b    c,d c,c
     ------------------------------------------------------------------------
                                  p                                 |
                                   a,d                              |
                                                                    |
                        l   l   p    + l   p                        |
                         b,c c,d b,b    b,d b,b                     |
                                                                    |
                  2                                                 |
                 l   l   p    + l   l   p    + l   p                |
                  b,c c,d b,b    b,c b,d b,b    c,d c,c             |
                                                                    |
      2   2                              2          2               |
     l   l   p    + 2l   l   l   p    + l   p    + l   p    + p     |
      b,c c,d b,b     b,c b,d c,d b,b    b,d b,b    c,d c,c    d,d  |

             4      4
o7 : Matrix R  <-- R

i8 : J = delete(0_R, flatten entries (L|W))

o8 = {p   , p   , l   , l   , p   , l   , p   , p   , p   }
       a,a   a,d   b,c   b,d   b,b   c,d   c,c   a,d   d,d

o8 : List

i9 : eliminate(J, ideal(S-M))

o9 = ideal (s   , s   )
             a,c   a,b

o9 : Ideal of R

i10 : gaussianVanishingIdeal(R)

o10 = ideal (s   , s   )
              a,c   a,b

o10 : Ideal of R

i11 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}})

o11 = MixedGraph{Bigraph => Bigraph{a => {d}}   }
                                    d => {a}
                 Digraph => Digraph{b => {c, d}}
                                    c => {d}
                                    d => {}
                 Graph => Graph{}

o11 : MixedGraph

i12 : R = gaussianRing G

o12 = R

o12 : PolynomialRing

i13 : M = gaussianParametrization(R,SimpleTreks=>true)

o13 = |    1          0                0               p          |
      |                                                 a,d       |
      |                                                           |
      |    0          1               l          l   l    + l     |
      |                                b,c        b,c c,d    b,d  |
      |                                                           |
      |    0         l                 1         l   l    + l     |
      |               b,c                         b,c b,d    c,d  |
      |                                                           |
      |  p     l   l    + l     l   l    + l            1         |
      |   a,d   b,c c,d    b,d   b,c b,d    c,d                   |

              4      4
o13 : Matrix R  <-- R

i14 : 
