Tropical And Idempotent Mathematics And Applications: International Workshop On Tropical And Idempotent Mathematics, August 26-31, 2012, Independent ... Moscow, Russia

616 Tropical and Idempotent Mathematics and Applications International Workshop Tropical and Idempotent Mathematics August 26–31, 2012 Independent University, Moscow, Russia G. L. Litvinov S. N. Sergeev Editors American Mathematical Society Tropical and Idempotent Mathematics and Applications International Workshop Tropical and Idempotent Mathematics August 26–31, 2012 Independent University, Moscow, Russia G. L. Litvinov S. N. Sergeev Editors 616 Tropical and Idempotent Mathematics and Applications International Workshop Tropical and Idempotent Mathematics August 26–31, 2012 Independent University, Moscow, Russia G. L. Litvinov S. N. Sergeev Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 15A80, 16Y60, 05C20, 14T05, 52A30, 35F21, 90C48, 65H20, 92C42, 82B30. Library of Congress Cataloging-in-Publication Data International Workshop on Tropical and Idempotent Mathematics (2012 : Moscow, Russia) Tropical and idempotent mathematics and applications : International Workshop on Tropical and Idempotent Mathematics, August 26–31, 2012, Independent University, Moscow, Russia / G.L. Litvinov, S.N. Sergeev, editors. pages cm – (Contemporary mathematics ; volume 616) Includes bibliographical references. ISBN 978-0-8218-9496-5 (alk. paper) 1. Tropical geometry–Congresses. 2. Geometry, Algebraic–Congresses. 3. Idempotents–Congresses. I. Litvinov, G. L. (Grigori˘ı Lazarevich), 1944– editor of compilation. II. Sergeev, S. N., 1981– editor of compilation. III. Title. QA582.I58 2012 516.35–dc23 2013041902 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/616 Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected] Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2014 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines  established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 19 18 17 16 15 14 Contents Preface vii In memory of Grigory Litvinov ix Tropical Cramer determinants revisited Marianne Akian, St´ ephane Gaubert, and Alexander Guterman 1 An approximation o