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KustinMiller :: stellarSubdivision

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

Synopsis

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**S. It is a subcomplex of the simplex on the variables of R**S.
i1 : R=QQ[x_0..x_4];
i2 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0);

o2 : Ideal of R
i3 : betti res I

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

o3 : BettiTally
i4 : D=idealToComplex(I)

o4 = {x  x  , x  x  , x  x  , x  x  , x  x  }
       2  4    0  3    0  2    1  3    1  4

o4 : complex with 5 facets on the vertices x  x  x  x  x  
                                            0  1  2  3  4
i5 : fc=facets D

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

o5 : List
i6 : S=QQ[x_5]

o6 = S

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

o7 = {x  x  , x  x  , x  x  , x  x  , x  x  , x  x  }
       5  4    5  2    0  3    0  2    1  3    1  4

o7 : complex with 6 facets on the vertices x  x  x  x  x  x  
                                            0  1  2  3  4  5
i8 : I5=complexToIdeal D5

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

o8 : Ideal of QQ[x , x , x , x , x , x ]
                  0   1   2   3   4   5
i9 : betti res 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=ideal(product((entries vars R)#0))

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

o11 : Ideal of R
i12 : D=idealToComplex I

o12 = {x  x  x  x  x  , x  x  x  x  x  , x  x  x  x  x  , x  x  x  x  x  , x 
        1  2  3  4  5    2  3  4  5  6    1  3  4  5  6    1  2  4  5  6    1
      -----------------------------------------------------------------------
      x  x  x  x  , x  x  x  x  x  }
       2  3  5  6    1  2  3  4  6

o12 : complex with 6 facets on the vertices x  x  x  x  x  x  
                                             1  2  3  4  5  6
i13 : S=QQ[x_7]

o13 = S

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

o14 = {x  x  x  x  x  , x  x  x  x  x  , x  x  x  x  x  , x  x  x  x  x  , x 
        4  5  7  2  3    4  5  7  1  3    4  5  7  1  2    2  3  4  5  6    1
      -----------------------------------------------------------------------
      x  x  x  x  , x  x  x  x  x  , x  x  x  x  x  , x  x  x  x  x  , x  x 
       3  4  5  6    1  2  4  5  6    5  6  7  2  3    5  6  7  1  3    5  6
      -----------------------------------------------------------------------
      x  x  x  , x  x  x  x  x  , x  x  x  x  x  , x  x  x  x  x  }
       7  1  2    4  6  7  2  3    4  6  7  1  3    4  6  7  1  2

o14 : complex with 12 facets on the vertices x  x  x  x  x  x  x  
                                              1  2  3  4  5  6  7
i15 : betti res complexToIdeal 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(Complex,Face,PolynomialRing)