Generalized splines on graphs with two labels and polynomial splines on cycles

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 Stanley$\unicode{x2013}$Stembridge 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 $G$ 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 $\mathbb{C}[x,y]$. 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.