regularTriangulationWeights -- height vector inducing a regular triangulation, if one exists

Usage:

regularTriangulationWeights T

regularTriangulationWeights(A, tri)
Inputs:
- T, an instance of the type Triangulation,
- A, a matrix,
- tri, a list,
Optional inputs:
- DegreeMatrix => a matrix, default value null, an integer matrix whose rows are a $\mathbb{Z}$-basis of $\ker A$; passed through to secondaryCone. Defaults to degreeMatrix T
- Strategy (missing documentation) => ..., default value Topcom,
Outputs:
- a list, of RR values (machine reals), one per column of the configuration: heights whose lower envelope yields tri; or null if the triangulation is not regular

Description

A triangulation is regular iff there is a height vector such that lifting each point to that height and taking the lower facets of the resulting upper hull recovers exactly the maximal simplices of the triangulation.

Computed via the engine LP rawConeInteriorPoint on the secondaryCone: an interior point $t \in \mathbb{R}^{N-d}$ is lifted back to weights $w \in \mathbb{R}^N$ via the Moore-Penrose pseudo-inverse of the charge matrix, $w = Q^\top (Q Q^\top)^{-1} t$. Any $w$ with $Q w = t$ induces the same triangulation; this lift is the canonical representative. For a rational/integer answer compatible with topcom's output, use topcomRegularTriangulationWeights.

i1 : A = transpose matrix {{0,3},{0,1},{-1,-1},{1,-1},{-4,-2},{4,-2}}

o1 = | 0 0 -1 1  -4 4  |
     | 3 1 -1 -1 -2 -2 |

              2       6
o1 : Matrix ZZ  <-- ZZ

i2 : T = regularFineTriangulation A

o2 = triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, {2, 3, 4}, {3, 4, 5}}

o2 : Triangulation

i3 : regularTriangulationWeights T

      97  47  1
o3 = {--, --, -, 0, 0, 0}
      16  16  8

o3 : List

i4 : isRegularTriangulation T

o4 = true

For a non-regular triangulation, the function returns null:

i5 : tri = {{0,1,2}, {1,3,5}, {2,3,4}, {0,1,5},
            {0,2,4}, {3,4,5}, {1,2,3}}

o5 = {{0, 1, 2}, {1, 3, 5}, {2, 3, 4}, {0, 1, 5}, {0, 2, 4}, {3, 4, 5}, {1,
     ------------------------------------------------------------------------
     2, 3}}

o5 : List

i6 : Tnr = triangulation(A, tri)

o6 = triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, {1, 2, 3}, {1, 3, 5}, {2, 3, 4}, {3, 4, 5}}

o6 : Triangulation

i7 : isRegularTriangulation Tnr

o7 = false

i8 : regularTriangulationWeights Tnr

Ways to use regularTriangulationWeights:

regularTriangulationWeights(Matrix,List)
regularTriangulationWeights(Triangulation)

For the programmer

The object regularTriangulationWeights is a method function with options.

The source of this document is in Triangulations.m2:1834:0.

regularTriangulationWeights -- height vector inducing a regular triangulation, if one exists

Description

See also

Ways to use regularTriangulationWeights:

For the programmer