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

## Synopsis

• Usage:
stellarSubdivision(D,F,S)
• Inputs:
• D, an object of class Complex, a simplicial complex on the variables of the polynomial ring R.
• F, an object of class Face, a face of D
• S, , a polynomial ring in one variable corresponding to the new vertex
• Outputs:
• an object of class Complex, 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=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