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stellarSubdivision -- Compute the stellar subdivision of a simplicial complex.

Description

Computes the stellar subdivision of a simplicial complex D by subdividing the face F with a new vertex corresponding to the variable of S. The result is a complex on the variables of R \otimes S. It is a subcomplex of the simplex on the variables of R \otimes S.

i1 : R=QQ[x_0..x_4];
 
i2 : I=monomialIdeal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0);

o2 : MonomialIdeal of R
 
i3 : betti freeResolution I

            0 1 2 3
o3 = total: 1 5 5 1
         0: 1 . . .
         1: . 5 5 .
         2: . . . 1

o3 : BettiTally
 
i4 : D=simplicialComplex I

o4 = simplicialComplex | x_2x_4 x_1x_4 x_1x_3 x_0x_3 x_0x_2 |

o4 : SimplicialComplex
 
i5 : fc=facets(D) / face

o5 = {x  x  , x  x  , x  x  , x  x  , x  x  }
       2  4    1  4    1  3    0  3    0  2

o5 : List
 
i6 : S=QQ[x_5]

o6 = S

o6 : PolynomialRing
 
i7 : D5=stellarSubdivision(D,fc#0,S)

o7 = simplicialComplex | x_4x_5 x_2x_5 x_1x_4 x_1x_3 x_0x_3 x_0x_2 |

o7 : SimplicialComplex
 
i8 : I5=ideal D5

o8 = ideal (x x , x x , x x , x x , x x , x x , x x , x x , x x )
             0 1   1 2   2 3   0 4   2 4   3 4   0 5   1 5   3 5

o8 : Ideal of QQ[x ..x ]
                  0   5
 
i9 : betti freeResolution I5

            0 1  2 3 4
o9 = total: 1 9 16 9 1
         0: 1 .  . . .
         1: . 9 16 9 .
         2: . .  . . 1

o9 : BettiTally
 

i10 : R=QQ[x_1..x_6]

o10 = R

o10 : PolynomialRing
 
i11 : I=monomialIdeal product gens R

o11 = monomialIdeal(x x x x x x )
                     1 2 3 4 5 6

o11 : MonomialIdeal of R
 
i12 : D=simplicialComplex I

o12 = simplicialComplex | x_2x_3x_4x_5x_6 x_1x_3x_4x_5x_6 x_1x_2x_4x_5x_6 x_1x_2x_3x_5x_6 x_1x_2x_3x_4x_6 x_1x_2x_3x_4x_5 |

o12 : SimplicialComplex
 
i13 : S=QQ[x_7]

o13 = S

o13 : PolynomialRing
 
i14 : Dsigma=stellarSubdivision(D,face {x_1,x_2,x_3},S)

o14 = simplicialComplex | x_2x_3x_5x_6x_7 x_1x_3x_5x_6x_7 x_1x_2x_5x_6x_7 x_2x_3x_4x_6x_7 x_1x_3x_4x_6x_7 x_1x_2x_4x_6x_7 x_2x_3x_4x_5x_7 x_1x_3x_4x_5x_7 x_1x_2x_4x_5x_7 x_2x_3x_4x_5x_6 x_1x_3x_4x_5x_6 x_1x_2x_4x_5x_6 |

o14 : SimplicialComplex
 
i15 : betti freeResolution ideal Dsigma

             0 1 2
o15 = total: 1 2 1
          0: 1 . .
          1: . . .
          2: . 1 .
          3: . 1 .
          4: . . .
          5: . . 1

o15 : BettiTally

See also

Ways to use stellarSubdivision:

  • stellarSubdivision(Fan,Matrix) -- see stellarSubdivision -- computes the stellar subdivision of the fan by a ray
  • stellarSubdivision(SimplicialComplex,Face,PolynomialRing)

For the programmer

The object stellarSubdivision is a method function.

The source of this document is in KustinMiller.m2:1133:0.