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arXiv · 2208.13062

Determinantal Conditions for Modules of Generalized Splines

Lauren L. Rose·Kariane Calta
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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.

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Lauren L. Rose, Kariane Calta. 2022-08-27. Determinantal Conditions for Modules of Generalized Splines. https://arxiv.org/abs/2208.13062

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