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Báo cáo toán học: "Riemann-Roch for Sub-Lattices of the Root Lattice An"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Riemann-Roch for Sub-Lattices of the Root Lattice An. | Riemann-Roch for Sub-Lattices of the Root Lattice An Omid Amini CNRS - DMA Ecole Normale Superieure Paris France oamini@math.ens.fr Madhusudan Manjunath Max-Planck Institut fur Informatik Saarbrucken Germany manjun@mpi-inf.mpg.de Submitted May 27 2009 Accepted Sep 1 2010 Published Sep 13 2010 Mathematics Subject Classification 05E99 52B20 52C07 05C38 Abstract 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. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R124 1 Contents 1 Introduction 3 2 Preliminaries 8 2.1 Sigma-Region of a Given Sub-lattice L of An. 8 2.2 Extremal Points of the Sigma-Region. 10 2.3 Min- and Max-Genus of Sub-Lattices of An and Uniform Lattices. 11 3 Proofs of Theorem 2.6 and Theorem 2.7 12 4 Voronoi Diagrams of Lattices under Simplicial Distance Functions 15 4.1 Polyhedral Distance Functions and their Voronoi Diagrams. 16 4.2 Voronoi Diagram of Sub-Lattices of An. 17 4.3 Vertices of Vor L that are Critical Points of a Distance Function. 20 4.4 Proof of Lemma 3.7. 23 5 Riemann-Roch Theorem for Uniform Reflection Invariant Sub-Lattices 24 5.1 A Riemann-Roch Inequality for Reflection Invariant Sub-Lattices Proof of Theorem 1.3. 24 5.2 Riemann-Roch Theorem for Uniform Reflection Invariant .