dim(Face,PolynomialRing) -- Compute the dimension of a face.

Synopsis

Function: dim
Usage:

dim(F,R)
Inputs:
- F, a face,
- R, a polynomial ring, with coker grading.
Outputs:
- an integer, bigger or equal to -1

Description

Computes the dimension of a face. If F.indices is present (usually the case by construction) this requires no computations.

If F.indices is not present but a polynomial ring R can be associated to F (which is the case if F.ofComplex is present (or given as a second argument) or F is non-empty) then R.grading (which can be installed by addCokerGrading) is used to compute the dimension of the plane spanned by F.

i1 : R=QQ[x_0..x_4]

o1 = R

o1 : PolynomialRing

i2 : addCokerGrading R

o2 = | -1 -1 -1 -1 |
     | 1  0  0  0  |
     | 0  1  0  0  |
     | 0  0  1  0  |
     | 0  0  0  1  |

              5       4
o2 : Matrix ZZ  <-- ZZ

i3 : C=simplex R

o3 = 4: x x x x x  
         0 1 2 3 4

o3 : complex of dim 4 embedded in dim 4 (printing facets)
     equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0

i4 : bC=boundaryOfPolytope C

o4 = 3: x x x x  x x x x  x x x x  x x x x  x x x x  
         0 1 2 3  0 1 2 4  0 1 3 4  0 2 3 4  1 2 3 4

o4 : complex of dim 3 embedded in dim 4 (printing facets)
     equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1

i5 : F=bC.fc_2_0

o5 = x x x
      0 1 2

o5 : face with 3 vertices

i6 : dim F

o6 = 2

Caveat

If F.indices is not present this returns a dimension as explained above but note that this does not check whether F is a face of the convex hull of the rows of R.grading.

Ways to use this method: