# getFacetBases -- given a slack matrix or a list of vertices of d-polytope or a rank d+1 matroid, or (d+1)-cone generators, creates a sorted list of vertices (empty if a matrix is given as input) in the order corresponding to B, and B the list of d spanning elements for each facet

## Synopsis

• Usage:
(newV, B) = getFacetBases S
(newV, B) = getFacetBases P
(newV, B) = getFacetBases M
(newV, B) = getFacetBases V
• Inputs:
• S, , a slack matrix S of rank d+1
• P, , a polytope
• M, , a matroid
• V, a list, a list of vertices of d-polytope, vectors of a rank d+1 matroid, or (d+1)-cone generators
• Optional inputs:
• Outputs:
• newV, a list, a list of vertices (empty if a matrix is given as input) in the order corresponding to B
• B, a list, list of d spanning elements for each facet

## Description

This function produces a list B of d spanning elements for each facet a given d-polytope or rank d+1 matroid, or (d+1)-cone generators. If a list of vertices is given as input, it also creates a sorted list of vertices from which B is computed

 `i1 : V = {{0, 0}, {1, 0}, {2, 1}, {1, 2}, {0, 1}};` ```i2 : (newV, B) = getFacetBases V o2 = ({{0, 0}, {1, 0}, {0, 1}, {2, 1}, {1, 2}}, {{0, 2}, {0, 1}, {1, 3}, {2, ------------------------------------------------------------------------ 4}, {3, 4}}) o2 : Sequence```
 `i3 : V = {{0, 0}, {1, 0}, {1, 1}, {0, 1}};` ```i4 : S = slackMatrix V; Order of vertices is {{0, 0}, {1, 0}, {0, 1}, {1, 1}} 4 4 o4 : Matrix QQ <--- QQ``` ```i5 : (newV, B) = getFacetBases S o5 = ({}, {{0, 2}, {1, 3}, {0, 1}, {2, 3}}) o5 : Sequence```

## See also

• slackFromPlucker -- given a slack matrix or a list of vertices of d-polytope or a rank d+1 matroid, or (d+1)-cone generators, it fills the corresponding slack matrix with Plucker coordinates
• symbolicSlackOfPlucker -- given the number of polytope vertices, cone generators, or matroid vectors, or a set of polytope vertices, cone generators, or matroid vectors, or a slack matrix and a set of set of hyperplane spanning set indices, it fills the slack matrix with Plucker variables
• grassmannSectionIdeal -- 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

## Ways to use getFacetBases :

• getFacetBases(List)
• getFacetBases(Matrix)
• getFacetBases(Matroid)
• getFacetBases(Polyhedron)