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@ Madhusudan Manjunath Max-Planck Institut fur Informatik Saarbrucken Germany manjun@ 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 Sigma-Region of a Given Sub-lattice L of An. 8 Extremal Points of the Sigma-Region. 10 Min- and Max-Genus of Sub-Lattices of An and Uniform Lattices. 11 3 Proofs of Theorem and Theorem 12 4 Voronoi Diagrams of Lattices under Simplicial Distance Functions 15 Polyhedral Distance Functions and their Voronoi Diagrams. 16 Voronoi Diagram of Sub-Lattices of An. 17 Vertices of Vor L that are Critical Points of a Distance Function. 20 Proof of Lemma . 23 5 Riemann-Roch Theorem for Uniform Reflection Invariant Sub-Lattices 24 A Riemann-Roch Inequality for Reflection Invariant Sub-Lattices Proof of Theorem . 24 Riemann-Roch Theorem for Uniform Reflection Invariant .
