Let R be a commutative ring with identity. An edge labeled graph is a graph with edges labeled by ideals of R. A generalized spline over an edge labeled graph is a vertex labeling by elements of R, such that the labels of any two adjacent vertices agree modulo the label associated to the edge connecting them. The set of generalized splines forms a subring and module over R. Such a module is called a generalized spline module. We show the existence of a flow-up basis for the generalized spline module on an edge labeled graph over a principal ideal domain by using a new method based on trails of the graph. We also give an algorithm to determine flow-up bases on arbitrary ordered cycles over any principal ideal domain.

中文翻译：

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任意图上广义样条模块的上流基

让R是一个具有恒等式的交换环。边缘标记图是具有由理想标记的边缘的图R. 边缘标记图上的广义样条是由元素标记的顶点R，使得任何两个相邻顶点的标签以与连接它们的边相关联的标签为模一致。广义样条集合形成一个子环和模块R. 这样的模块称为广义样条模块。我们通过使用一种基于图轨迹的新方法，在主要理想域上的边缘标记图上展示了广义样条模块的上流基础的存在。我们还给出了一种算法来确定基于任意主要理想域上的任意有序循环的上流。