AlgebraicSplines -- a package for working with splines on simplicial complexes, polytopal complexes, and graphs

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 ℝⁿ. The spline module S^r(Δ) is the module of all functions f continuously differentiable to order r such that f is a polynomial when restricted to each facet σ∈Δ. The vector space S^r_d(Δ) consists of splines whose polynomial restrictions have degree at most d. Its dimension is of particular interest in approximation theory (see [10]).

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 dimS^r_d(Δ)=dimS^r(cΔ)_d, where S^r(cΔ)_d is the vector space of splines of degree exactly d on the cone cΔ over Δ. So the statistic dimS^r_d(Δ) is the Hilbert function of S^r(cΔ). The functions splineMatrix and splineModule construct the matrix from [3] whose kernel is S^r(Δ), and the module S^r(Δ), respectively. The functions splineDimensionTable and hilbertComparisonTable display the statistic dimS^r_d(Δ) and compare it to the long term dimension formula, respectively.

In [1], Billera uses a homological approach to solve a conjecture of Strang on the dimension of the space S¹_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 dimS^r_d(Δ); perhaps most notable is [11]. In [10] the chain complex is also used to compute dimension formulas in the polytopal setting. The Billera-Schenck-Stillman complex is implemented by the method splineComplex. The Billera-Schenck-Stillman chain complex is a quotient of the cellular chain complex of Δ (relative to the boundary of Δ); the latter is computed by cellularComplex. The kernel of this surjection is given by idealsComplex.

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 S^r(Δ) as a quotient of a polynomial ring, thus recovering its ring structure. In [2], Billera shows that there is a natural isomorphism between the affine Stanley-Reisner ring K_a[Δ] (see explanation in documentation for ringStructure) of a simplicial complex and the ring S⁰(Δ) of continuous piecewise polynomials on Δ. The isomorphism identifies the variable corresponding to a vertex with the Courant function for that vertex. The method courantFunctions computes these functions, and the method stanleyReisner constructs the isomorphism identified in [2]. Moreover, the rings S^r(Δ) live inside S⁰(Δ) - this is of particular interest when Δ is simplicial due to Billera’s result. The method stanleyReisnerPresentation constructs S^r(Δ) as a quotient of a ring map from a polynomial ring into S⁰(Δ); in particular the generators of S^r(Δ) are presented in terms of generators of S⁰(Δ). If Δ is simplicial these are selected to be the Courant functions.

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.