Basis Criteria for Extending Generalized Splines

Let $R$ be a commutative ring with identity and $G$ a graph. Extending generalized splines are a further extension of generalized splines by allowing vertex labels of $G$ to lie in varying modules rather than in a fixed ring $R$. Geometrically, this corresponds to the construction of equivariant cohomology by Braden and MacPherson (see [5]). Therefore, characterizing such splines has immediate implications in geometry, particularly in the computation of equivariant cohomology. In this paper, we study extending generalized splines as a $R$- module in which each vertex $v$ is labeled by $M_v = m_v R$ and each edge $e$ is labeled by $M_e = R/r_e R$ together with quotient $R$-module homomorphisms $M_v\to M_e$ for each vertex $v$ incident to the edge $e$, where $R$ is a greatest common divisor domain (GCD). We characterize module bases of such splines in terms of determinants so that it provides a criterion for freeness of spline modules.

💡 Research Summary

The paper investigates “extending generalized splines,” a broadening of the generalized spline framework in which each vertex of a graph is labeled by a possibly different module rather than a single fixed ring. Specifically, the authors consider a commutative ring (R) that is a greatest common divisor (GCD) domain, assign to each vertex (v) the module (M_v = m_v R) (an ideal of (R)), and to each edge (e) the quotient module (M_e = R / r_e R). For every incident vertex–edge pair they use the natural quotient homomorphism, and a spline is a collection of vertex elements satisfying the compatibility condition (\varphi_u(f_u)=\varphi_v(f_v)) across each edge (uv). The set of all such labelings, denoted (\widehat R_G), forms an (R)-module.

The central contribution is a determinant‑based criterion for when a collection of splines forms a basis of (\widehat R_G). To formulate this, the authors introduce a combinatorial invariant (\widehat Q_G). For each vertex (v_i) they consider all “trails” (edge‑simple paths) ending at (v_i); for each trail they take the greatest common divisor of the edge labels along the trail. Then they define \