i1 : loadPackage "SimplicialComplexes";
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The pentagonal bipyramid has 7 vertices, 15 edges and 10 triangles.
i2 : R = ZZ[a..g];
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i3 : bipyramid = simplicialComplex monomialIdeal(
a*g, b*d, b*e, c*e, c*f, d*f)
o3 = | efg bfg deg cdg bcg aef abf ade acd abc |
o3 : SimplicialComplex
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i4 : f = fVector bipyramid
o4 = HashTable{-1 => 1}
0 => 7
1 => 15
2 => 10
o4 : HashTable
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i5 : f#0
o5 = 7
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i6 : f#1
o6 = 15
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i7 : f#2
o7 = 10
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Every simplicial complex other than the void complex has a unique face of dimension -1.
i8 : void = simplicialComplex monomialIdeal 1_R
o8 = 0
o8 : SimplicialComplex
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i9 : fVector void
o9 = HashTable{-1 => 0}
o9 : HashTable
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For a larger examp;le we consider the polarization of an artinian monomial ideal from section 3.2 in Miller-Sturmfels, Combinatorial Commutative Algebra.
i10 : S = ZZ[x_1..x_4, y_1..y_4, z_1..z_4];
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i11 : 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);
o11 : MonomialIdeal of S
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i12 : D = simplicialComplex I;
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i13 : fVector D
o13 = HashTable{-1 => 1 }
0 => 12
1 => 66
2 => 220
3 => 492
4 => 768
5 => 837
6 => 609
7 => 264
8 => 51
o13 : HashTable
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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.