Chordal -- exploiting chordal structure in polynomial ideals

i1 : R = QQ[a,b,c,d];

i2 : I = ideal {a^2-1, a^2+a*b+1, a^3+c^2, b*d + d, c^3+c*d};

o2 : Ideal of R

i3 : G = constraintGraph I

o3 = Graph{a => {b, c}}
           b => {a, d}
           c => {a, d}
           d => {b, c}

o3 : Graph

i4 : Gc = chordalGraph G

o4 = ChordalGraph{a => {b, c} }
                  b => {c, d}
                  c => {d}
                  d => {}

o4 : ChordalGraph

i5 : I = chromaticIdeal(QQ, cycleGraph 10, 3);

o5 : Ideal of QQ[a..j]

i6 : N = chordalNet I;

i7 : chordalElim N;

i8 : N

                                     2    2   2          2
o8 = ChordalNet{ a => {{a*b - a*j + b  - j , a  + a*j + j }} }
                        2          2
                 b => {b  + b*c + c }
                        2          2
                 c => {c  + c*d + d }
                        2          2
                 d => {d  + d*e + e }
                        2          2
                 e => {e  + e*f + f }
                        2          2
                 f => {f  + f*g + g }
                        2          2
                 g => {g  + g*h + h }
                        2                2
                 h => {h  + h*i - i*j - j }
                        2          2
                 i => {i  + i*j + j }
                        3
                 j => {j  - 1}

o8 : ChordalNet

i9 : sum for f in gbList I list #terms f

o9 = 253

i10 : I = adjacentMinorsIdeal(QQ,2,10);

o10 : Ideal of QQ[a..t]

i11 : N = chordalNet I;

i12 : chordalTria N;

i13 : N

o13 = ChordalNet{ a => { , a*d - b*c}                     }
                  b => { , b,  }
                  c => {c,  ,  , c*f - d*e}
                  d => {d, d,  ,  }
                  e => { ,  , e*h - f*g, e,  , e*h - f*g}
                  f => { , f,  , f,  ,  }
                  g => {g,  ,  , g*j - h*i,  , g*j - h*i}
                  h => {h, h,  ,  ,  ,  }
                  i => { ,  , i*l - j*k, i,  , i*l - j*k}
                  j => { , j,  , j,  ,  }
                  k => {k,  ,  , k*n - l*m,  , k*n - l*m}
                  l => {l, l,  ,  ,  ,  }
                  m => { ,  , m*p - n*o, m,  , m*p - n*o}
                  n => { , n,  , n,  ,  }
                  o => {o,  ,  , o*r - p*q,  , o*r - p*q}
                  p => {p, p,  ,  ,  ,  }
                  q => { ,  , q*t - r*s, q,  , q*t - r*s}
                  r => { , r,  , r,  ,  }
                  s => {s,  ,  ,  }
                  t => {t,  ,  }

o13 : ChordalNet

i14 : dim N

o14 = 11

i15 : topComponents N

i16 : codimCount N

         9
o16 = 55t

o16 : ZZ[t]