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

- 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) => a Boolean value, default value true,
  - CoefficientRing (missing documentation) => a ring, default value QQ,
  - VariableName (missing documentation) => a symbol, default value t,
  - InputType (missing documentation) => a string, default value "ByFacets",
- Outputs:
  - M, a matrix, 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.

## See also

- 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.
- ringStructure -- given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
- 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 $\Delta$.

## Ways to use courantFunctions:

- `courantFunctions(List,List)`

## For the programmer

The object courantFunctions is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/AlgebraicSplines.m2:2173:0.