# courantFunctions -- returns the Courant functions of a simplicial complex

## Synopsis

• Usage:
M=courantFunctions(V,F)
• Inputs:
• V, a list, a list of vertex coordinates
• F, a list, a list of facets, recorded as indices of V
• Optional inputs:
• BaseRing (missing documentation) => a ring, default value null,
• Homogenize (missing documentation) => , default value true,
• CoefficientRing (missing documentation) => a ring, default value QQ,
• VariableName (missing documentation) => , default value t,
• InputType (missing documentation) => , default value ByFacets,
• Outputs:
• M, , a matrix so that column $i$ is the Courant function corresponding to vertex $V_i$

## Description

This method returns a matrix with as many rows as facets and as many columns as vertices. Column $i$ of the matrix is the Courant function corresponding to vertex $V_i$. This is the piecewise linear function which takes the value $1$ at the vertex $V_i$ and $0$ at all other vertices.

 i1 : V={{0,0},{0,1},{-1,-1},{1,0}}; i2 : F={{0,1,2},{0,2,3},{0,1,3}}; i3 : courantFunctions(V,F) o3 = | 2t_0-t_1+t_2 -t_0+t_1 -t_0 0 | | -t_0+2t_1+t_2 0 -t_1 t_0-t_1 | | -t_0-t_1+t_2 t_1 0 t_0 | 3 4 o3 : Matrix (QQ[t ..t ]) <--- (QQ[t ..t ]) 0 2 0 2

If the option Homogenize=>false is given, the Courant function corresponding to a cone vertex (if there is one) will be discarded.

 i4 : S=QQ[x,y]; i5 : courantFunctions(V,F,Homogenize=>false,BaseRing=>S) o5 = | -x+y -x 0 | | 0 -y x-y | | y 0 x | 3 3 o5 : Matrix S <--- S

The Courant functions are used to construct the Stanley Reisner ring of a simplicial complex. See stanleyReisner.

• stanleyReisner -- Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned.
• stanleyReisnerPresentation -- creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$.