2025/11/29 by Dilaver, Gökçen, Selma Altınok, Altinok, Selma
Engineering · Computer Science · Mathematics · #Advanced Numerical Analysis Techniques #Polynomial and algebraic computation #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2512.00429
Let R be a commutative ring with identity and G a graph. An extending generalized spline on G is a vertex labeling f ∈ ∏v Mv, where for each edge e=uv there exists an R-module Muv together with homomorphisms φu : Mu → Muv and φv : Mv → Muv such that φu(fu)=φv(fv). Extending generalized splines are further generalizations for generalized splines. They can also be considered as generalized splines over modules. In this paper, we prove that some of the results for splines can be extended to generalized splines over modules Mv=mv\mathbb Z at each vertex v and we define a method of a graph reduction based on graph operations on vertices and edges to produce an explicit ℤ-module basis for generalized splines over modules. This corresponds to a sequence of surjective homomorphisms between the associated spline modules so that the space of splines decomposes as a direct sum of certain submodules.