Generalized splines on graphs with two labels and polynomial splines on cycles
Portia Anderson, Jacob P. Matherne, Julianna Tymoczko

TL;DR
This paper develops a generalized algebraic framework for splines on graphs with two labels, providing new generators and algorithms applicable to cycles and other graphs, unifying concepts across combinatorics, algebra, and topology.
Contribution
It introduces a comprehensive theory of generalized splines on graphs with specific edge-labeling conditions, including explicit generators and algorithms for their construction.
Findings
Generated minimal sets for all graphs with two finitely-generated ideals as edge labels.
Produced generators for cycles with edge labels as squares of linear forms in polynomial rings.
Provided a constructive, computer-implementable algorithm for computing generalized splines.
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 former 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 StanleyStembridge conjecture, 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 with each edge labeled by an ideal in a ring consists of a vertex-labeling by elements of so that the labels…
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Digital Image Processing Techniques · Polynomial and algebraic computation
