The pentagonal bipyramid has 7 vertices, 15 edges
and 10 triangles.
i1 : R = ZZ[a..g];
|
i2 : bipyramid = simplicialComplex monomialIdeal(
a*g, b*d, b*e, c*e, c*f, d*f)
o2 = | efg bfg deg cdg bcg aef abf ade acd abc |
o2 : SimplicialComplex
|
i3 : f = fVector bipyramid
o3 = HashTable{-1 => 1}
0 => 7
1 => 15
2 => 10
o3 : HashTable
|
i4 : f#0
o4 = 7
|
i5 : f#1
o5 = 15
|
i6 : f#2
o6 = 10
|
Every simplicial complex other than the void
complex has a unique face of dimension -1.
i7 : void = simplicialComplex monomialIdeal 1_R
o7 = 0
o7 : SimplicialComplex
|
i8 : fVector void
o8 = HashTable{-1 => 0}
o8 : HashTable
|
For a larger examp;le we consider the polarization
of an artinian monomial ideal from section 3.2 in
Miller-Sturmfels, Combinatorial Commutative Algebra.
i9 : S = ZZ[x_1..x_4, y_1..y_4, z_1..z_4];
|
i10 : I = monomialIdeal(x_1*x_2*x_3*x_4,
y_1*y_2*y_3*y_4,
z_1*z_2*z_3*z_4,
x_1*x_2*x_3*y_1*y_2*z_1,
x_1*y_1*y_2*y_3*z_1*z_2,
x_1*x_2*y_1*z_1*z_2*z_3);
o10 : MonomialIdeal of S
|
i11 : D = simplicialComplex I;
|
i12 : fVector D
o12 = HashTable{-1 => 1 }
0 => 12
1 => 66
2 => 220
3 => 492
4 => 768
5 => 837
6 => 609
7 => 264
8 => 51
o12 : HashTable
|
The f-vector is computed using the Hilbert series
of the Stanley-Reisner ideal. For example, see
Hosten and Smith's
chapter Monomial Ideals, in Computations in
Algebraic Geometry with Macaulay2, Springer 2001.