33 research outputs found
Generalized Splines on Arbitrary Graphs
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 (algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky, Kottwitz, and MacPherson, among others). The central question of this paper asks when an arbitrary edge-labeled graph has nontrivial generalized splines. The answer is “always”, and we prove the stronger result that the module of generalized splines contains a free submodule whose rank is the number of vertices in G. We describe the module of 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
Associated Primes of Spline Complexes
The spline complex whose top homology is
the algebra of mixed splines over the fan
was introduced by Schenck-Stillman in
[Schenck-Stillman 97] as a variant of a complex
of Billera [Billera 88]. In this paper we
analyze the associated primes of homology modules of this complex. In
particular, we show that all such primes are linear. We give two applications
to computations of dimensions. The first is a computation of the third
coefficient of the Hilbert polynomial of , including cases
where vanishing is imposed along arbitrary codimension one faces of the
boundary of , generalizing the computations in [Geramita-Schenck
98,McDonald-Schenck 09]. The second is a description of the fourth coefficient
of the Hilbert polynomial of for simplicial fans
. We use this to derive the result of Alfeld, Schumaker, and Whiteley
on the generic dimension of tetrahedral splines for
[Alfeld-Schumaker-Whiteley 93] and indicate via an example how this description
may be used to give the fourth coefficient in particular nongeneric
configurations.Comment: 40 pages, 10 figure