# 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, , 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 polytope
• C, , a cone
• M, , 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 .