Dimension of splines on graphs in the case of degree two and smoothness one in two variables

TL;DR

The paper resolves the degree-2, smoothness-1 spline-dimension problem on planar graphs with edge labels $(ax+by+c)^2$ by recasting spline conditions as linear constraints via the extended cycle-basis matrix $M^{\mathsf{ext}}$ and introducing a contractibility framework. It develops edge-injective functions, generic edge labels, and a homogenization technique to reduce to homogeneous cases, proving that the dimension equals $e_G-\operatorname{rank}(M^{\mathsf{ext}})$ and providing a concrete algorithm to compute it. The authors show that, under no proper contractible subset of faces, a generic edge labeling exists, yielding $\operatorname{rank}(M^{\mathsf{ext}})=3f_G$ (hence $\dim\mathsf{Spl}_2(G,\ell;v_0)=e_G-3f_G$); in the minimal-contractible case, $\operatorname{rank}(M^{\mathsf{ext}})=e_G$ and the dimension collapses to $0$. They also describe how determinants decompose via edge-injective functions and contractions, characterize special-position edge-label loci, and present an algorithmic route to the dimension computation, with extensions to triangulations and discussion of open questions. This yields a precise, combinatorial, and algorithmic framework for understanding spline spaces on graphs in the stated degree/smoothness regime.

Abstract

Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data interpolation, to create smooth curves in computer graphics and to find numerical solutions to partial differential equations. Gilbert, Tymoczko, and Viel generalized the classical splines combinatorially and algebraically: a generalized spline is a vertex labeling of a graph $G$ by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the ideal generated by the corresponding edge label. We study the generalized splines on the planar graphs whose edges are labeled by two-variable polynomials of the form $(ax+by+c)^2$ and whose vertices are labeled by polynomials of degree at most two. In this paper we address the upper-bound conjecture for the dimension of degree-2 splines of smoothness 1. The dimension is expressed in terms of the rank of the extended cycle basis matrix. We also provide a combinatorial algorithm on graphs to compute the rank by contracting certain subgraphs.

Figures (13)

• Figure 1: Example of spline given in Example \ref{['example.graph']}. Center: Graph $G=(V,E)$ with edges labeled $e_1,\ldots,e_8$. Left: Edge-labeled graph $(G,\ell)$ with edge labels indicated in red. Right: A spline with the function $p$ on the vertices indicated in blue.
• Figure 2: A triangulation describing a particular system of splines, the corresponding (non-homogeneous) edge-labeling of its dual graph, and the associated homogeneous edge-labeled graph.
• Figure 3: The maps $\varphi_{\ell'}$ and $\varphi_{\ell_0}$ send the splines on the left and right, respectively, to the scalars labeling edges in the center as referred to in Example \ref{['example: continuing homogeneous and nonhom example.']}
• Figure 4: Triangulation and its dual graph which appeared in MS.1975. This triangulation will be further analyzed in Example \ref{['example: Morgan-Scott']}.
• Figure 5: Two different planar embeddings of the same graph.

Theorems & Definitions (107)

• Definition 1.1
• Example 1.2
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• Remark 2.6
• Example 2.7
• Lemma 2.8