Generalized Splines over $\mathbb{Z}$-Modules on Arbitrary Graphs

TL;DR

The paper extends spline theory to extending generalized splines labeled by Z-modules on arbitrary graphs, introducing a robust graph-reduction framework to construct explicit Z-module bases. It defines the module-theoretic spline space Ř_G with vertex labels M_v = m_v Z and edge labels M_uv = Z/r_uv Z, and develops flow-up bases and CRT-based techniques to analyze subgraphs and reductions. A key contribution is the establishment of surjective maps between spline modules and a direct-sum decomposition Ř_G ≅ Ř_Gred ⊕ ker ψ, enabling iterative reduction to a single vertex and the explicit assembly of a basis from flow-up elements. The results generalize prior ring-based spline theory to module-valued vertex data and provide concrete methods for basis construction and structural analysis of spline modules on graphs, with potential implications for algebraic graph theory and related applications.

Abstract

Let $R$ be a commutative ring with identity and $G$ a graph. An extending generalized spline on $G$ is a vertex labeling $f \in \prod_{v} M_v$, where for each edge $e=uv$ there exists an $R$-module $M_{uv}$ together with homomorphisms $ \varphi_u : M_u \to M_{uv}$ and $ \varphi_v : M_v \to M_{uv}$ such that $\varphi_u(f_u)=\varphi_v(f_v).$ Extending generalized splines are further generalizations for generalized splines. They can also be considered as generalized splines over modules. In this paper, we prove that some of the results for splines can be extended to generalized splines over modules $M_v=m_v\mathbb Z$ at each vertex $v$ and we define a method of a graph reduction based on graph operations on vertices and edges to produce an explicit $\mathbb{Z}$-module basis for generalized splines over modules. This corresponds to a sequence of surjective homomorphisms between the associated spline modules so that the space of splines decomposes as a direct sum of certain submodules.

Figures (7)

• Figure 1: An edge-labeled path graph $(P_4,\beta)$
• Figure 2: An edge-labeled complete graph $(K_4,\beta)$
• Figure 3: A vertex reduced graph $(G_{v_4},\beta_{v_4})$ on $v_4$
• Figure 4: An edge-labeled reduced graph $(G_{red},\beta_{red})$
• Figure 5: An edge-labeled cycle graph $(C_n,\beta)$

Theorems & Definitions (48)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Theorem 2.9: The Chinese Remainder Theorem
• Example 2.10