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Báo cáo toán học: "How to Draw Tropical Planes"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: How to Draw Tropical Planes. | How to Draw Tropical Planes Sven Herrmann Department of Mathematics Technische Universitat Darmstadt Germany sherrmann@mathemat ik.tu-darmstadt.de Michael Joswigi Department of Mathematics Technische Universitat Darmstadt Germany j oswig@mathematik.tu-darmstadt.de Anders Jensen Courant Research Center Georg-August-Universitat Gottingen Germany j ensen@uni-math.gwdg.de Bernd Sturmfels Department of Mathematics University of California Berkeley USA bernd@math.berkeley.edu Submitted Sep 1 2008 Accepted Apr 14 2009 Published Apr 20 2009 Mathematics Subject Classification 52B40 14M15 05C05 Dedicated to Anders Bjorner on the occasion of his 60th birthday. Abstract The tropical Grassmannian parameterizes tropicalizations of ordinary linear spaces while the Dressian parameterizes all tropical linear spaces in TPn-1. We study these parameter spaces and we compute them explicitly for n 7. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are then used to draw pictures. 1 Introduction A line in tropical projective space TPn-1 is an embedded metric tree which is balanced and has n unbounded edges pointing into the coordinate directions. The parameter space of these objects is the tropical Grassmannian Gr 2 n . This is a simplicial fan 29 known to evolutionary biologists as the space of phylogenetic trees with n labeled leaves 24 3.5 and known to algebraic geometers as the moduli space of rational tropical curves 23 . This author was supported by a Graduate Grant of TU Darmstadt. iThis author was supported by a Sofia Kovalevskaja prize awarded to Olga Holtz at TU Berlin. This author was supported by the DFG Research Unit Polyhedral Surfaces . This author was supported by an Alexander-von-Humboldt senior award at TU Berlin and the US National Science Foundation. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R6 1 Speyer 27 28 introduced higher-dimensional tropical linear spaces. They are contractible polyhedral complexes all