Basis Criteria for Extending Generalized Splines
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918322699763712 |
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| author | Dilaver, Gökçen Altınok, Selma |
| author_facet | Dilaver, Gökçen Altınok, Selma |
| contents | Let $R$ be a commutative ring with identity and $G$ a graph. Extending generalized splines are a further extension of generalized splines by allowing vertex labels of $G$ to lie in varying modules rather than in a fixed ring $R$. Geometrically, this corresponds to the construction of equivariant cohomology by Braden and MacPherson (see [5]). Therefore, characterizing such splines has immediate implications in geometry, particularly in the computation of equivariant cohomology.
In this paper, we study extending generalized splines as a $R$- module in which each vertex $v$ is labeled by $M_v = m_v R$ and each edge $e$ is labeled by $M_e = R/r_e R$ together with quotient $R$-module homomorphisms $M_v\to M_e$ for each vertex $v$ incident to the edge $e$, where $R$ is a greatest common divisor domain (GCD). We characterize module bases of such splines in terms of determinants so that it provides a criterion for freeness of spline modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04440 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Basis Criteria for Extending Generalized Splines Dilaver, Gökçen Altınok, Selma Combinatorics 05C78, 05C25, 05C50, 11A07, 13C05 Let $R$ be a commutative ring with identity and $G$ a graph. Extending generalized splines are a further extension of generalized splines by allowing vertex labels of $G$ to lie in varying modules rather than in a fixed ring $R$. Geometrically, this corresponds to the construction of equivariant cohomology by Braden and MacPherson (see [5]). Therefore, characterizing such splines has immediate implications in geometry, particularly in the computation of equivariant cohomology. In this paper, we study extending generalized splines as a $R$- module in which each vertex $v$ is labeled by $M_v = m_v R$ and each edge $e$ is labeled by $M_e = R/r_e R$ together with quotient $R$-module homomorphisms $M_v\to M_e$ for each vertex $v$ incident to the edge $e$, where $R$ is a greatest common divisor domain (GCD). We characterize module bases of such splines in terms of determinants so that it provides a criterion for freeness of spline modules. |
| title | Basis Criteria for Extending Generalized Splines |
| topic | Combinatorics 05C78, 05C25, 05C50, 11A07, 13C05 |
| url | https://arxiv.org/abs/2602.04440 |