Document Type
Article
Publication Date
1-26-2024
Publication Title
The Electronic Journal of Combinatorics
Abstract
Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The for- mer consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley–Stembridge con- jecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet. This paper focuses on the theory of generalized splines. A generalized spline on a graph G with each edge labeled by an ideal in a ring R consists of a vertex-labeling by elements of R so that the labels on adjacent vertices u, v differ by an element of the ideal associated to the edge uv. We study the R-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: 1) for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and 2) for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form (ax + by)2 in the polynomial ring C[x, y]. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines.
Volume
31
Issue
1
DOI
https://doi.org/10.37236/11155
Creative Commons License
This work is licensed under a Creative Commons Attribution-No Derivative Works 4.0 International License.
Rights
© The Authors
Recommended Citation
Anderson, Portia; Matherne, Jacob P.; and Tymoczko, Julianna, "Generalized Splines on Graphs with Two Labels and Polynomial Splines on Cycles" (2024). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/184
Comments
Archived as publsihed.