grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix

Synopsis

Usage:

I = grassmannSectionIdeal(S, B)

I = grassmannSectionIdeal S

I = grassmannSectionIdeal(V, B)

I = grassmannSectionIdeal V

I = grassmannSectionIdeal P

I = grassmannSectionIdeal C

I = grassmannSectionIdeal M
Inputs:
- S, a matrix, slack matrix
- B, a list, set of hyperplane spanning set indices
- V, a list, list of polytope vertex coordinates, cone generators, or matroid vectors
- P, a convex polyhedron, a polytope
- C, a convex rational cone, a cone
- M, a matroid, a matroid
Optional inputs:
- CoefficientRing => ..., default value QQ, specifies the coefficient ring of the underlying ring of the ideal
- Object => ..., default value "polytope", specify combinatorial object
- Saturate => ..., default value "all", specifies saturation strategy to be used
- Strategy => ..., default value Eliminate, specifies saturation strategy to be used
Outputs:
- I, an ideal, the Grassmannian section ideal corresponding to choice B of the object with slack matrix S

Description

Given a slack matrix of a polytope, a cone or a matroid, or a set of polytope vertices, cone generators, or matroid vectors, and a set of set of hyperplane spanning set indices, it computes the Grassmannian section ideal corresponding to choice B of the object with slack matrix S.

i1 : V = {{0, 0}, {1, 0}, {2, 1}, {1, 2}, {0, 1}};

i2 : (VV, B) = getFacetBases V;

i3 : I = grassmannSectionIdeal(VV, B)

Order of vertices is 
{{0, 0}, {1, 0}, {0, 1}, {2, 1}, {1, 2}}

o3 = ideal (p     p      - p     p      + p     p     , p     p      -
             1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4  
     ------------------------------------------------------------------------
     p     p      + p     p     , p     p      - p     p      + p     p     ,
      0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4 
     ------------------------------------------------------------------------
     p     p      - p     p      + p     p     , p     p      - p     p     
      1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4
     ------------------------------------------------------------------------
     + p     p     )
        0,1,2 0,3,4

o3 : Ideal of QQ[p     ..p     , p     , p     , p     , p     , p     , p     , p     , p     ]
                  0,1,2   0,1,3   0,2,3   1,2,3   0,1,4   0,2,4   1,2,4   0,3,4   1,3,4   2,3,4

i4 : V = {{0, 0}, {1, 0}, {2, 1}, {1, 2}, {0, 1}};

i5 : (VV, B) = getFacetBases V;

i6 : I = grassmannSectionIdeal(slackMatrix(VV), B)

Order of vertices is 
{{0, 0}, {1, 0}, {0, 1}, {2, 1}, {1, 2}}

o6 = ideal (p     p      - p     p      + p     p     , p     p      -
             1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4  
     ------------------------------------------------------------------------
     p     p      + p     p     , p     p      - p     p      + p     p     ,
      0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4 
     ------------------------------------------------------------------------
     p     p      - p     p      + p     p     , p     p      - p     p     
      1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4
     ------------------------------------------------------------------------
     + p     p     )
        0,1,2 0,3,4

o6 : Ideal of QQ[p     ..p     , p     , p     , p     , p     , p     , p     , p     , p     ]
                  0,1,2   0,1,3   0,2,3   1,2,3   0,1,4   0,2,4   1,2,4   0,3,4   1,3,4   2,3,4

Ways to use grassmannSectionIdeal :

grassmannSectionIdeal(Cone)
grassmannSectionIdeal(List)
grassmannSectionIdeal(List,List)
grassmannSectionIdeal(Matrix)
grassmannSectionIdeal(Matrix,List)
grassmannSectionIdeal(Matroid)
grassmannSectionIdeal(Polyhedron)

For the programmer

The object grassmannSectionIdeal is a method function with options.

grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix

Synopsis

Description

See also

Ways to use grassmannSectionIdeal :

For the programmer