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# simplexComplex(ZZ,PolynomialRing) -- make the simplex as an abstract simplicial complex

## Synopsis

• Function: simplexComplex
• Usage:
simplexComplex (d, S)
• Inputs:
• d, an integer, that is the dimension of the simplex
• S, , that has at least $d+1$ generators
• Outputs:
• , that has a unique facet of dimension $d$

## Description

A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. The $d$-simplex is the unique $d$-dimensional abstract simplicial complex having one facet. Furthermore, in the $d$-simplex, there are $\binom{d+1}{k+1}$ faces having dimension $k$.

 i1 : S = ZZ[a..e]; i2 : irrelevant = simplexComplex (-1, S) o2 = simplicialComplex | 1 | o2 : SimplicialComplex i3 : monomialIdeal irrelevant o3 = monomialIdeal (a, b, c, d, e) o3 : MonomialIdeal of S i4 : dim irrelevant o4 = -1 i5 : fVector irrelevant o5 = {1} o5 : List i6 : assert(facets irrelevant === {1_S}) i7 : assert(dim irrelevant === -1 and fVector irrelevant === {1})
 i8 : Δ0 = simplexComplex (0, S) o8 = simplicialComplex | a | o8 : SimplicialComplex i9 : monomialIdeal Δ0 o9 = monomialIdeal (b, c, d, e) o9 : MonomialIdeal of S i10 : dim Δ0 o10 = 0 i11 : fVector Δ0 o11 = {1, 1} o11 : List i12 : assert(facets Δ0 === {a} and dim Δ0 === 0) i13 : assert(fVector Δ0 == {1,1})
 i14 : Δ1 = simplexComplex (1, S) o14 = simplicialComplex | ab | o14 : SimplicialComplex i15 : monomialIdeal Δ1 o15 = monomialIdeal (c, d, e) o15 : MonomialIdeal of S i16 : dim Δ1 o16 = 1 i17 : fVector Δ1 o17 = {1, 2, 1} o17 : List i18 : assert(facets Δ1 === {a*b} and dim Δ1 === 1) i19 : assert(fVector Δ1 === {1,2,1})
 i20 : Δ2 = simplexComplex (2, S) o20 = simplicialComplex | abc | o20 : SimplicialComplex i21 : monomialIdeal Δ2 o21 = monomialIdeal (d, e) o21 : MonomialIdeal of S i22 : dim Δ2 o22 = 2 i23 : fVector Δ2 o23 = {1, 3, 3, 1} o23 : List i24 : assert(facets Δ2 === {a*b*c} and dim Δ2 === 2) i25 : assert(fVector Δ2 === {1,3,3,1})
 i26 : Δ3 = simplexComplex (3, S) o26 = simplicialComplex | abcd | o26 : SimplicialComplex i27 : monomialIdeal Δ3 o27 = monomialIdeal e o27 : MonomialIdeal of S i28 : dim Δ3 o28 = 3 i29 : fVector Δ3 o29 = {1, 4, 6, 4, 1} o29 : List i30 : assert(facets Δ3 === {a*b*c*d} and dim Δ3 === 3) i31 : assert(fVector Δ3 === toList apply(-1..3, i -> binomial(3+1,i+1)))
 i32 : Δ4 = simplexComplex (4, S) o32 = simplicialComplex | abcde | o32 : SimplicialComplex i33 : monomialIdeal Δ4 o33 = monomialIdeal () o33 : MonomialIdeal of S i34 : dim Δ4 o34 = 4 i35 : fVector Δ4 o35 = {1, 5, 10, 10, 5, 1} o35 : List i36 : assert(facets Δ4 === {a*b*c*d*e} and dim Δ4 === 4) i37 : assert(fVector Δ4 === toList apply(-1..4, i -> binomial(4+1,i+1)))

The vertices in the $d$-simplex are the first $d+1$ variables in the given polynomial ring.