Basis Criteria for Generalized Spline Modules via Determinant

Abstract

Given a graph whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling by the elements of R such that the difference of the labels on adjacent vertices lies in the ideal associated to the edge. The set of generalized splines has a ring and an R-module structure. We study the module structure of generalized splines where the base ring is a greatest common divisor domain. We give basis criteria for generalized splines on cycles, diamond graphs and trees by using determinantal techniques. In the last section of the paper, we define a graded module structure for generalized splines and give some applications of the basis criteria for cycles, diamond graphs and trees.

arXiv comment20 pages, 10 figures

References in corpus (6)

• Generalized splines on arbitrary graphs
• Integer Generalized Splines on Cycles
• Bases and Structure Constants of Generalized Splines with Integer Coefficients on Cycles
• Flow-up Bases for Generalized Spline Modules on Arbitrary Graphs
• Splines over integer quotient rings
• Splines mod m

Cited by in corpus (1)

• Graphs admitting only constant splines