Tropical Algebraic Geometry
Tropical geometry is algebraic geometry over the semifield of tropical numbers, i.e., the real numbers and negative infinity enhanced with the (max,+)-arithmetics. Geometrically, tropical varieties are much simpler than their classical counterparts. Yet t
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Ilia Itenberg Grigory Mikhalkin Eugenii Shustin
Tropical Algebraic Geometry
Birkhäuser Basel · Boston · Berlin
Authors: Ilia Itenberg IRMA, Université Louis Pasteur 7 rue René Descartes 67084 Strasbourg Cedex France e-mail: [email protected]
Eugenii Shustin School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Ramat Aviv, 69978 Tel Aviv Israel e-mail: [email protected]
Grigory Mikhalkin Department of Mathematics University of Toronto Toronto, Ont. M5S 2E4 Canada e-mail: [email protected]
2000 Mathematics Subject Classification 14M25, 14N35, 14N10, 52B20, 14P25, 14H99
Library of Congress Control Number: 2009923622
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 978-3-0346-0047-7 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. First edition 2007
© 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-0346-0047-7
e-ISBN 978-3-0346-0048-4
987654321
www.birkhauser.ch
Contents Preface
vii
Preface to the second edition
ix
1 Introduction to tropical geometry 1.1 Images under the logarithm . . . . . . . . . 1.2 Families of amoebas . . . . . . . . . . . . . 1.3 Non-Archimedean amoebas . . . . . . . . . 1.4 Non-standard complex numbers . . . . . . . 1.5 The tropical semifield T . . . . . . . . . . . 1.6 Tropical curves and integer affine structure 1.7 Exercises . . . . . . . . . . . . . . . . . . .
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2 Patchworking of algebraic varieties 2.1 Introduction: A general idea of the patchworking construction . . . 2.2 Elements of toric geometry . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Construction of toric varieties . . . . . . . . . . . . . . . . . 2.2.2 A toric variety associated with a fan . . . . . . . . . . . . . 2.2.3 A toric variety associated with a convex lattice polyhedron 2.2.4 Embedding of Tor(Δ) into a projective space . . . . . . . . 2.2.5 The real part of a toric variety and the moment map . . . . 2.2.6 Hypersurfaces in toric varieties . . . . . . . . . . . . . . . . 2.3 Viro’s patchworking method . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Chart of a real polynomial . . . . . . . . . . . . . . . . . . . 2.3.2 Patchworking of real nonsingu
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