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A generalized spline on an edge labeled graph $(G,\alpha)$ is defined as avertex labeling, such that the difference of labels on adjacent vertices liesin the ideal generated by the edge label. We study generalized splines overgreatest common divisor domains and present a determinantal basis condition forgeneralized spline modules on arbitrary graphs. The main result of the paperanswers a conjecture that appeared in several papers.

Basis condition for generalized spline modules