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Extending Generalized Splines Over The Integers

Gökçen Dilaver, Selma Altınok

TL;DR

The paper extends generalized splines to vertex-labeled $\\mathbb{Z}$-modules and develops a comprehensive framework, including an extended GKM matrix, to study the $R$-module structure of the spline space $\\hat{R}_G$. It provides a path-graph construction for $M_v = m_v \\mathbb{Z}$ and a general longest-path technique to build flow-up bases on arbitrary graphs, establishing existence and basis properties via divisibility conditions and CRT-enabled solvability. Key contributions include explicit first-nonzero-entry formulas for flow-up classes on paths, a basis condition for $\\hat{R}_G$ over $\\mathbb{Z}$, and a constructive, scalable method to obtain a $\\mathbb{Z}$-module basis for spline spaces on general graphs (including reductions for zero-labeled vertices). Together these results broaden spline theory to module-valued vertex labels and enable explicit computation using kernel/syzygy approaches. The work has potential applications in integer-valued spline theory and modular-structure modeling, providing both theoretical insight and practical algorithms for basis construction.

Abstract

Let $R$ be a commutative ring with identity and $G$ a graph. \emph{An extending generalized spline} on $G$ is a vertex labeling $f \in \prod_{v} M_v$ such that at each edge $e=uv$ there is an $R$-module $M_{uv}$ together with homomorphisms $ \varphi_u : M_u \to M_{uv}$ and $ \varphi_v : M_v \to M_{uv}$ for each vertex $u, v$ incident to the edge $e$ so 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. The main goal of this paper is to study the $R$-module structure of extending generalized splines. We concentrate on two following questions: which of the results for general splines extend to generalized splines over modules and if there is an algorithm or an explicit formula for special basis classes, called a flow up basis, for generalized splines over modules. We show that certain results concerning generalized splines can be extended to a setting where each vertex $v$ is assigned a module $M_v=m_v\mathbb Z$. We provide an algorithm to construct a special basis for generalized splines over these modules on paths. Additionally, we introduce a new technique to construct a flow-up basis on arbitrary graphs using the idea of an algorithm on paths.

Extending Generalized Splines Over The Integers

TL;DR

The paper extends generalized splines to vertex-labeled -modules and develops a comprehensive framework, including an extended GKM matrix, to study the -module structure of the spline space . It provides a path-graph construction for and a general longest-path technique to build flow-up bases on arbitrary graphs, establishing existence and basis properties via divisibility conditions and CRT-enabled solvability. Key contributions include explicit first-nonzero-entry formulas for flow-up classes on paths, a basis condition for over , and a constructive, scalable method to obtain a -module basis for spline spaces on general graphs (including reductions for zero-labeled vertices). Together these results broaden spline theory to module-valued vertex labels and enable explicit computation using kernel/syzygy approaches. The work has potential applications in integer-valued spline theory and modular-structure modeling, providing both theoretical insight and practical algorithms for basis construction.

Abstract

Let be a commutative ring with identity and a graph. \emph{An extending generalized spline} on is a vertex labeling such that at each edge there is an -module together with homomorphisms and for each vertex incident to the edge so that Extending generalized splines are further generalizations for generalized splines. They can also be considered as generalized splines over modules. The main goal of this paper is to study the -module structure of extending generalized splines. We concentrate on two following questions: which of the results for general splines extend to generalized splines over modules and if there is an algorithm or an explicit formula for special basis classes, called a flow up basis, for generalized splines over modules. We show that certain results concerning generalized splines can be extended to a setting where each vertex is assigned a module . We provide an algorithm to construct a special basis for generalized splines over these modules on paths. Additionally, we introduce a new technique to construct a flow-up basis on arbitrary graphs using the idea of an algorithm on paths.
Paper Structure (6 sections, 20 theorems, 87 equations, 17 figures)

This paper contains 6 sections, 20 theorems, 87 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Background and Notations
  3. The Matrix $M_{G}$
  4. A Basis Condition over $\mathbb{Z}$-module
  5. A Flow-up Class for Path Graphs over $\mathbb{Z}$
  6. Longest Paths Technique

Key Result

Lemma 2.5

Let $(G,\beta)$ be an edge-labeled graph with a $\mathbb{Z}$-module $M_v= m_v \mathbb{Z}$ at each vertex where $\beta(uv)= \mathbb{Z}/r_{uv} \mathbb{Z} \subset \mathbb{Z}$ at each edge $uv$. Then there exists a nontrivial spline on $(G,\beta)$.

Figures (17)

  • Figure 1: An edge-labeled path graph $P_2$
  • Figure 2: An edge-labeled path graph $P_2$
  • Figure 3: An edge-labeled path graph $P_3$
  • Figure 4: An edge-labeled path graph $(P_2,\beta)$
  • Figure 5: An edge-labeled path graph $(P_3,\beta)$
  • ...and 12 more figures

Theorems & Definitions (63)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5: Existence of non-trivial splines
  • ...and 53 more