Generalized splines on graphs with two labels and polynomial splines on cycles
share
A generalized spline on a graph G with edges labeled by ideals 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 edge-labelings consist of at most two finitely-generated ideals, and 2) for cycles when the edge-labelings consist 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 classical (analytic) splines.
READ FULL TEXT