Abstract
Let [Formula: see text] be a commutative ring with identity. An edge labeled graph is a graph with edges labeled by ideals of [Formula: see text]. A generalized spline over an edge labeled graph is a vertex labeling by elements of [Formula: see text], 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 [Formula: see text]. 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.
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