Basis condition for generalized spline modules

TL;DR

This paper establishes a determinantal basis condition for generalized spline modules on arbitrary graphs over GCD domains, resolving a previously posed conjecture in the field.

Contribution

It introduces a new basis criterion for generalized splines on graphs over GCD domains, providing a significant theoretical advancement.

Findings

Derived a determinantal basis condition for generalized spline modules

Proved the conjecture related to spline modules on arbitrary graphs

Extended the theory of splines to GCD domains

Abstract

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