Universality theorems for generalized splines
摘要
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.
引用
@article{arxiv.2605.24348,
title = {Universality theorems for generalized splines},
author = {Jacob Matherne and Eric Ramos and Julianna Tymoczko},
journal= {arXiv preprint arXiv:2605.24348},
year = {2026}
}