2017/05/31 by McCleary Philbin, Lindsay Swift, Philbin, McCleary +5
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1706.00105
openalex publication_date 2017/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a graph with edges labeled by elements in ℤ/mℤ, a generalized spline is a labeling of each vertex by an integer \mod m such that the labels of adjacent vertices agree modulo the label associated to the edge connecting them. These generalize the classical splines that arise in analysis as well as in a construction of equivariant cohomology often referred to as GKM-theory. We give an algorithm to produce minimum generating sets for the ℤ-module of splines on connected graphs over ℤ/m ℤ. As an application, we give a quick heuristic to determine the minimum number of generators of the module of splines over ℤ/mℤ. We also completely determine the ring of splines over ℤ/pkℤ by providing explicit multiplication tables with respect to the elements of our minimum generating set. Our final result extends some of these results to splines over ℤ.