Splines over integer quotient rings

TL;DR

This paper develops algorithms and explicit descriptions for splines over integer quotient rings, extending classical and equivariant cohomology concepts to modular settings and providing tools for understanding their algebraic structure.

Contribution

It introduces an algorithm for minimal generating sets of splines over rac{m}{rac{Z}} and explicitly characterizes the ring structure over rac{p^k}{rac{Z}}, extending to splines over rac{Z}.

Findings

Provided an algorithm for minimal generating sets of splines.

Explicit multiplication tables for splines over rac{p^k}{rac{Z}}.

Extended results to splines over rac{Z}.

Abstract

Given a graph with edges labeled by elements in $\mathbb{Z}/m\mathbb{Z}$, 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 $\mathbb{Z}$-module of splines on connected graphs over $\mathbb{Z}/m \mathbb{Z}$. As an application, we give a quick heuristic to determine the minimum number of generators of the module of splines over $\mathbb{Z}/m\mathbb{Z}$. We also completely determine the ring of splines over $\mathbb{Z}/p^k\mathbb{Z}$ 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 $\mathbb{Z}$.