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# splineModule -- compute the module of all splines on partition of a space

## Synopsis

• Usage:
M = splineModule(V,F,E,r)
M = splineModule(V,F,r)
• Inputs:
• V, a list, list of coordinates of vertices of $\Delta$
• F, a list, list of facets of $\Delta$; each facet is recorded as a list of indices of vertices taken from V
• E, a list, list of codimension one faces of $\Delta$ (interior or not); each codimension one face is recorded as a list of indices of vertices taken from V
• r, an integer, desired degree of smoothness
• 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, , module of splines on $\Delta$

## Description

This method returns the spline module. It is presented as the image of a matrix whose columns generate splines as a module over the polynomial ring. Each column represents a spline whose entries are the polynomials restricted to the facets of the complex $\Delta$.

 i1 : V = {{0,0},{1,0},{1,1},{0,1}} o1 = {{0, 0}, {1, 0}, {1, 1}, {0, 1}} o1 : List i2 : F = {{0,1,2},{0,2,3}} o2 = {{0, 1, 2}, {0, 2, 3}} o2 : List i3 : E = {{0,1},{0,2},{0,3},{1,2},{2,3}} o3 = {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {2, 3}} o3 : List i4 : splineModule(V,F,E,1) o4 = image | 1 t_0^2-2t_0t_1+t_1^2 | | 1 0 | 2 o4 : QQ[t ..t ]-module, submodule of (QQ[t ..t ]) 0 2 0 2

• splineMatrix -- compute matrix whose kernel is the module of $C^r$ splines on $\Delta$