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arXiv · 2605.24348

Universality theorems for generalized splines

Jacob Matherne·Eric Ramos·Julianna Tymoczko
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Abstract

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating set'' for the module of splines over any graph with fixed combinatorial genus. This theorem holds over any Noetherian commutative ring with a chosen finite list of ideals for edge-labels. We then give several applications of this theorem, including showing that a particular generating function associated to splines on trees is algebraic when the base ring satisfies certain finiteness conditions. We illustrate our technical theorems explicitly by giving a classification of splines on graphs with combinatorial genus one and two.

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Jacob Matherne, Eric Ramos, Julianna Tymoczko. 2026-05-23. Universality theorems for generalized splines. https://arxiv.org/abs/2605.24348

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