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      Splines over integer quotient rings

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          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 \(\mathbb{Z}\).

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          Generalized splines on arbitrary graphs

          , , (2015)
          Let G be a graph whose edges are labeled by ideals of a commutative ring. We introduce a generalized spline, which is a vertex-labeling of G by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the corresponding edge ideal. Generalized splines arise naturally in combinatorics (em algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky-Kottwitz-MacPherson and others). The central question of this manuscript asks when an arbitrary edge-labeled graph has nontrivial generalized splines. The answer is `always', and we prove the stronger result that generalized splines contain a free submodule whose rank is the number of vertices in G. We describe all generalized splines when G is a tree, and give several ways to describe the ring of generalized splines as an intersection of generalized splines for simpler subgraphs of G. We also present a new tool which we call the GKM matrix, an analogue of the incidence matrix of a graph, and end with open questions.
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            Journal
            2017-05-31
            Article
            1706.00105
            570bc1a4-f6df-4534-862b-c4203df44b2f

            http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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            math.CO

            Combinatorics
            Combinatorics

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