Abstract
We study {\em generalized graph splines,} introduced by Gilbert, Viel, and the last author. For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.
Highlights
This paper studies generalized splines, which are parametrized by a ring, a graph, and a map from the edges of the graph to ideals in the ring
We focus on the R-module structure of SR(G, α) in this paper
We study operations that induce a decomposition of the module of splines
Summary
This paper studies generalized splines, which are parametrized by a ring, a graph, and a map from the edges of the graph to ideals in the ring. Suppose (G, α) is a connected graph whose set of edge labels can be linearly ordered by inclusion. This follows directly from the previous theorem once we note that the graph consisting of the edge ab is generated as an R-module by the identity spline together with the spline that is β on b and 0 on a and that χB is the extension of this latter spline to all of G.
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