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

## Synopsis

• Usage:
stellarSubdivision(D,F,S)
• Inputs:
• D, , a simplicial complex on the variables of the polynomial ring R.
• F, an object of class Face (missing documentation), a face of D
• S, , a polynomial ring in one variable corresponding to the new vertex
• Outputs:
• , the stellar subdivision of D with respect to F and S

## 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=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 res 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 = | x_2x_4 x_1x_4 x_1x_3 x_0x_3 x_0x_2 | o4 : SimplicialComplex``` ```i5 : fc=facets(D,useFaceClass=>true) 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 = | 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 , 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=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 = | 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 = | 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 res ideal Dsigma 0 1 2 o15 = total: 1 2 1 0: 1 . . 1: . . . 2: . 1 . 3: . 1 . 4: . . . 5: . . 1 o15 : BettiTally```