Determinantal Conditions for Modules of Generalized Splines

TL;DR

This paper investigates conditions under which the module of generalized splines on a graph over a ring is free, providing determinantal criteria for the existence of a basis and flow-up class basis.

Contribution

It introduces determinantal conditions that characterize when the spline module is free and when a flow-up class basis exists, extending understanding of spline modules over rings.

Findings

Determinantal conditions for module freeness

Criteria for existence of flow-up class basis

Extension of spline theory to general rings

Abstract

Generalized splines on a graph $G$ with edge labels in a commutative ring $R$ are vertex labelings such that if two vertices share an edge in $G$, the difference between the vertex labels lies in the ideal generated by the edge label. When $R$ is an integral domain, the set of all such splines is a finitely generated $R$-module $R_G$ of rank $n$, the number of vertices of $G$. We find determinantal conditions on subsets of $R_G$ that determine whether $R_G$ is a free module, and if so, whether a so called "flow-up class basis" exists.