This package provides methods for computations with piecewise polynomial functions (splines) over polytopal complexes.

Let *Δ* be a partition (simplicial,polytopal,cellular,rectilinear, etc.) of a space *ℝ ^{n}*. The spline module

The foundations of the algebraic approach to splines (as it applies to numerical analysis) was developed by Billera and Rose in [1],[2],[3]. In particular, it is shown in [3] that dim*S ^{r}_{d}(Δ)=*dim

In [1], Billera uses a homological approach to solve a conjecture of Strang on the dimension of the space *S ^{1}_{d}(Δ)*. The chain complex which he defines in [1] was later modified by Schenck and Stillman in [10]; we will call this the Billera-Schenck-Stillman chain complex. It has appeared in many papers due to its use in finding dim

The functions mentioned thus far are concerned only with the structure of *S ^{r}(Δ)* as a module over the polynomial ring. The method ringStructure constructs

In topology, splines arise as equivariant cohomology of spaces with a torus action via GKM theory - see [14] for a survey of how this relates to splines in numerical analysis; and [6] for the precise relationship between continuous splines and the equivariant Chow cohomology of toric varieties. From this perspective, the notion of generalized splines on graphs was introduced in [5]. The method generalizedSplines computes splines in this more flexible setting. The relationship to splines in numerical analysis is via the dual graph described by Rose in [7],[8].

Additionally, there are connections between splines and the module of multi-derivations of a hyperplane arrangement. A basic structural connection was noticed in [3]. For the braid arrangement and its sub-arrangements, the module of derivations is isomorphic to a ring of splines in a natural way (see [11] and [4]).

Methods in this package borrow from code written by Hal Schenck.

References:

[1] Louis J. Billera. Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. Amer. Math. Soc., 310(1):325–340, 1988.

[2] Louis J. Billera, The Algebra of Continuous Piecewise Polynomials, Adv. in Math. 76, 170-183 (1989).

[3] Louis J. Billera and Lauren L. Rose. A dimension series for multivariate splines. Discrete Comput. Geom., 6(2):107–128, 1991.

[4] Michael DiPasquale. Generalized Splines and Graphic Arrangements. J. Algebraic Combin. 45 (2017), no. 1, 171-189.

[5] Simcha Gilbert, Julianna Tymoczko, and Shira Viel. Generalized splines on arbitrary graphs. Pacific J. Math. 281 (2016), no. 2, 333-364.

[6] Sam Payne, Equivariant Chow cohomology of toric varieties, Math. Res. Lett. 13 (2006), 29-41.

[7] Lauren Rose, Combinatorial and topological invariants of modules of piecewise polynomials, Adv. Math. 116 (1995), 34-45.

[8] Lauren Rose, Graphs, syzygies, and multivariate splines, Discrete Comput. Geom. 32 (2004), 623-637.

[9] T. McDonald, H. Schenck, Piecewise polynomials on polyhedral complexes, Adv. in Appl. Math. 42 (2009), 82-93.

[10] Hal Schenck and Mike Stillman. Local cohomology of bivariate splines. J. Pure Appl. Algebra,117/118:535–548, 1997. Algorithms for algebra (Eindhoven, 1996).

[11] Hal Schenck, A Spectral Sequence for Splines, Adv. in Appl. Math. 19, 183-199 (1997).

[12] Schenck, Hal . Splines on the Alfeld split of a simplex and type A root systems. J. Approx. Theory 182 (2014), 1-6.

[13] Gilbert Strang, Piecewise Polynomials and the Finite Element Method, Bull. Amer. Math. Soc. 79 (1973) 1128-1137.

[14] Julianna Tymoczko. Splines in geometry and topology. Comput. Aided Geom. Design 45 (2016), 32-47.

- Functions and commands
- cellularComplex -- create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary
- courantFunctions -- returns the Courant functions of a simplicial complex
- formsList -- list of powers of (affine) linear forms cutting out a specified list of codimension one faces.
- generalizedSplines -- the module of generalized splines associated to a simple graph with an edge labelling
- hilbertComparisonTable -- a table to compare the values of the hilbertFunction and hilbertPolynomial of a graded module
- idealsComplex -- creates the Billera-Schenck-Stillman chain complex of ideals
- postulationNumber -- computes the largest degree at which the hilbert function of the graded module M is not equal to the hilbertPolynomial
- 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
- splineComplex -- creates the Billera-Schenck-Stillman chain complex
- splineDimensionTable -- a table with the dimensions of the graded pieces of a graded module
- splineMatrix -- compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
- splineModule -- compute the module of all splines on partition of a space
- 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 $\Delta$.

- Symbols
- RingType, see generalizedSplines -- the module of generalized splines associated to a simple graph with an edge labelling
- GenVar, see 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
- IdempotentVar, see 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
- Trim, see 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
- VariableGens, see 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
- BaseRing, see splineMatrix -- compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
- ByFacets, see splineMatrix -- compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
- ByLinearForms, see splineMatrix -- compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
- Homogenize, see splineMatrix -- compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
- InputType, see splineMatrix -- compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
- VariableName, see splineMatrix -- compute matrix whose kernel is the module of $C^r$ splines on $\Delta$