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 $\mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in classical (analytic) splines.

翻译：以美元环形理想标注边缘的G$平面图上的通用样板由按美元元素标出的顶点标签组成,以美元为单位,使相邻的顶点的标签因与边缘相关的理想要素的一个要素而异。我们研究了通用样条的R$模块,为几个组的图形和边缘标签制作了最小生成数据集:当边缘标签包含最多两个有限生成的理想时,所有图点的顶点标签为1美元;当边缘标签包含以美元(x+by)形式元素生成的主要理想时,周期为2美元。我们用一种适合计算机实施的建设性算法获取发电机,并提供了多种应用,包括在经典(分析性)样条纹中根据背景得出的若干结果。