Graphs, Syzygies, and Multivariate Splines (English)

In: Discrete & Computational Geometry   ;  32 ,  4  ;  623-637  ;  2004

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Abstract The module of splines on a polyhedral complex can be viewed as the syzygy module of its dual graph with edges weighted by powers of linear forms. When the assignment of linear forms to edges meets certain conditions, we can decompose the graph into disjoint cycles without changing the isomorphism class of the syzygy module. Thus we can use this decomposition to compute the homological dimension and the Hilbert series of the module. We provide alternate proofs of some results of Schenck and Stillman, extending those results to the polyhedral case. We also provide examples which illustrate the role that geometry plays in determining the syzygy module.

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Guest Editors#8217 Preface
Bayer, Margaret M. / Lee, Carl W. / Sturmfels, Bernd | 2004
Guest Editors’ Preface
Bayer, Margaret M. / Lee, Carl W. / Sturmfels, Bernd | 2004
The Partial-Fractions Method for Counting Solutions to Integral Linear Systems
Beck, Matthias | 2004
Isoradial Bodies
Brandenberg, René / Dattasharma, Abhi / Gritzmann, Peter / Larman, David | 2004
Vertices of Gelfand--Tsetlin Polytopes
Loera, Jesús A. / McAllister, Tyrrell B. | 2004
A Complexity Bound on Faces of the Hull Complex
Develin, Mike | 2004
Enumerative Properties of Ferrers Graphs
Ehrenborg, Richard / Willigenburg, Stephanie | 2004
Tchebyshev Posets
Hetyei, Gábor | 2004
Mixed Fibre Polytopes
McMullen, Peter | 2004
Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices
Nyman, Kathryn / Swartz, Ed | 2004
Convex Combinatorial Optimization
Onn, Shmuel / Rothblum, Uriel G. | 2004
Non-Crossing Frameworks with Non-Crossing Reciprocals
Orden, David / Rote, Günter / Santos, Francisco / Servatius, Brigitte / Servatius, Herman / Whiteley, Walter | 2004
The Et-Construction for Lattices, Spheres and Polytopes
Paffenholz, Andreas / Ziegler, Günter M. | 2004
Graphs, Syzygies, and Multivariate Splines
Rose, Lauren L. | 2004