2 research outputs found
Basis Criteria for Generalized Spline Modules via Determinant
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.Comment: 20 pages, 10 figure
Flow-up Bases for Generalized Spline Modules on Arbitrary Graphs
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 it 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