Riemann-Roch for Sub-Lattices of the Root Lattice A(n)

作者：Amini Omid^*; Manjunath Madhusudan

来源：Electronic Journal of Combinatorics, 2010, 17(1).

Recently, Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph G. In this paper, we develop a general Riemann-Roch theory for sub-lattices of the root lattice An analogue to the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we obtain a geometric proof of the Riemann-Roch theorem for graphs and generalise the result to other sub-lattices of An. In particular, we provide a new geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.

• 出版日期2010-9-13