Explicit deformation of lattice ideals via chip-firing games on directed graphs

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• 作者：Spencer Backman ; Madhusudan Manjunath
• 关键词：Lattice ideals ; Binomial ideals ; Chip ; firing on graphs ; Directed graphs ; Laplacian matrix ; Scarf complex ; Free resolutions of ideals ; Explicit deformation
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：42
• 期：4
• 页码：1097-1110
• 全文大小：509 KB
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• 作者单位：Spencer Backman (1)
  Madhusudan Manjunath (2)
  
  1. Department of Computer Science, University of Rome 鈥淟a Sapienza鈥? Rome, Italy
  2. Department of Mathematics, University of California, Berkeley, CA, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Order, Lattices and Ordered Algebraic Structures
  Computer Science, general
  Group Theory and Generalizations
  
• 出版者：Springer U.S.
• ISSN：1572-9192

文摘

For a finite index sublattice L of the root lattice of type A, we construct a deterministic algorithm to deform the lattice ideal \(I_\mathrm{L}\) to a nearby generic lattice ideal, answering a question posed by Miller and Sturmfels. Our algorithm is based on recent results of Perkinson, Perlman, and Wilmes concerning commutative algebraic aspects of chip-firing on directed graphs. As an application of our deformation algorithm, we construct a cellular resolution of the lattice ideal \(I_\mathrm{L}\) by degenerating the Scarf complex of its deformation. Keywords Lattice ideals Binomial ideals Chip-firing on graphs Directed graphs Laplacian matrix Scarf complex Free resolutions of ideals Explicit deformation