loadPackage "SimplicialComplexes"; |
The pentagonal bipyramid has 7 vertices, 15 edges and 10 triangles.
R = ZZ[a..g]; |
bipyramid = simplicialComplex monomialIdeal(
a*g, b*d, b*e, c*e, c*f, d*f) |
f = fVector bipyramid |
f#0 |
f#1 |
f#2 |
Every simplicial complex other than the void complex has a unique face of dimension -1.
void = simplicialComplex monomialIdeal 1_R |
fVector void |
For a larger examp;le we consider the polarization of an artinian monomial ideal from section 3.2 in Miller-Sturmfels, Combinatorial Commutative Algebra.
S = ZZ[x_1..x_4, y_1..y_4, z_1..z_4]; |
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); |
D = simplicialComplex I; |
fVector D |
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.