Universality theorems for generalized splines
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916041457664000 |
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| author | Matherne, Jacob Ramos, Eric Tymoczko, Julianna |
| author_facet | Matherne, Jacob Ramos, Eric Tymoczko, Julianna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24348 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Universality theorems for generalized splines Matherne, Jacob Ramos, Eric Tymoczko, Julianna Combinatorics Category Theory Rings and Algebras Representation Theory 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. |
| title | Universality theorems for generalized splines |
| topic | Combinatorics Category Theory Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2605.24348 |