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Basis Criteria for Extending Generalized Splines

Gökçen Dilaver, Selma Altınok

TL;DR

This work addresses the problem of characterizing bases for extending generalized spline modules $\\hat{R}_G$ over a GCD domain $R$, where vertex labels are $M_v = m_vR$ and edge labels are $M_e = R/r_eR$. It introduces a determinant-based criterion built from the trail-based quantity $\\hat{Q}_G$ and shows that, when $\\lvert F_1,\\dots,F_n\\rvert = u\\hat{Q}_G$ with a unit $u\\in R$, the set is an $R$-basis; in the PID case this criterion is necessary and sufficient and bases can be realized as flow-up bases with rank $n$. The paper also discusses limitations in non-PID GCD domains, provides examples where flow-up bases fail to exist, and connects the framework to the geometric interpretation of equivariant cohomology. Overall, it offers a practical determinantal test for freeness and constructive basis results for extending generalized spline modules, with potential impact on computations in equivariant topology and related graph-algebra settings.

Abstract

Let $R$ be a commutative ring with identity and $G$ a graph. Extending generalized splines are a further extension of generalized splines by allowing vertex labels of $G$ to lie in varying modules rather than in a fixed ring $R$. Geometrically, this corresponds to the construction of equivariant cohomology by Braden and MacPherson (see [5]). Therefore, characterizing such splines has immediate implications in geometry, particularly in the computation of equivariant cohomology. In this paper, we study extending generalized splines as a $R$- module in which each vertex $v$ is labeled by $M_v = m_v R$ and each edge $e$ is labeled by $M_e = R/r_e R$ together with quotient $R$-module homomorphisms $M_v\to M_e$ for each vertex $v$ incident to the edge $e$, where $R$ is a greatest common divisor domain (GCD). We characterize module bases of such splines in terms of determinants so that it provides a criterion for freeness of spline modules.

Basis Criteria for Extending Generalized Splines

TL;DR

This work addresses the problem of characterizing bases for extending generalized spline modules over a GCD domain , where vertex labels are and edge labels are . It introduces a determinant-based criterion built from the trail-based quantity and shows that, when with a unit , the set is an -basis; in the PID case this criterion is necessary and sufficient and bases can be realized as flow-up bases with rank . The paper also discusses limitations in non-PID GCD domains, provides examples where flow-up bases fail to exist, and connects the framework to the geometric interpretation of equivariant cohomology. Overall, it offers a practical determinantal test for freeness and constructive basis results for extending generalized spline modules, with potential impact on computations in equivariant topology and related graph-algebra settings.

Abstract

Let be a commutative ring with identity and a graph. Extending generalized splines are a further extension of generalized splines by allowing vertex labels of to lie in varying modules rather than in a fixed ring . Geometrically, this corresponds to the construction of equivariant cohomology by Braden and MacPherson (see [5]). Therefore, characterizing such splines has immediate implications in geometry, particularly in the computation of equivariant cohomology. In this paper, we study extending generalized splines as a - module in which each vertex is labeled by and each edge is labeled by together with quotient -module homomorphisms for each vertex incident to the edge , where is a greatest common divisor domain (GCD). We characterize module bases of such splines in terms of determinants so that it provides a criterion for freeness of spline modules.
Paper Structure (4 sections, 16 theorems, 61 equations, 5 figures)

This paper contains 4 sections, 16 theorems, 61 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and Notations
  3. Determinant-Based Conditions on Edge-Labeled Graphs
  4. Determinantal Criteria Over PID

Key Result

Theorem 1.1

Let $(G,\beta)$ be an edge-labeled graph and $R$ a GCD. Suppose $F_1, F_2, \ldots, F_n \in \hat{R}_G$ are splines. If the determinant $\lvert F_1, F_2, \ldots, F_n \rvert = u \hat{Q}_G$ for some unit $u \in R$, then the set $\{F_1, F_2, \ldots, F_n\}$ forms an $R$-module basis of $\hat{R}_G$.

Figures (5)

  • Figure 1: An edge-labeled path graph $P_2$.
  • Figure 2: An edge-labeled cycle graph $(C_3,\beta)$
  • Figure 3: Tree graph $T_4$
  • Figure 4: An edge-labeled graph $(G,\beta)$
  • Figure 5: An edge-labeled cycle graph $(C_3,\beta)$

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 31 more