Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface

Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface JGeomAnal https://doi.org/10.1007/s12220-018-0041-3 Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface 1 2 Mounir Nisse · Timur Sadykov Received: 17 November 2017 © Mathematica Josephina, Inc. 2018 Abstract Given a complex algebraic hypersurface H , we introduce a subset of the Newton polytope of the defining polynomial for H which is a polyhedral complex and enjoys the key topological and combinatorial properties of the amoeba of H for a large class of hypersurfaces. We provide an explicit formula for this polyhedral complex in the case when the spine of the amoeba is dual to a triangulation of the Newton polytope of the defining polynomial. In particular, this yields a description of the polyhedral complex when the hypersurface is optimal (Forsberg et al. in Adv Math 151:45–70, 2000). We conjecture that a polyhedral complex with these properties exists in general. Keywords Amoebas · Newton polytope · Tropical geometry · Polyhedral complex Mathematics Subject Classification 32A60 · 52B55 The presented research has been performed in the framework of the basic part of the scientific research state task in the field of scientific activity of the Ministry of Education and Science of the Russian Federation, Project No. 2.9577.2017/8.9. B Timur Sadykov Sadykov.TM@rea.ru Mounir Nisse http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Geometric Analysis Springer Journals

Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface

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Publisher
Springer US
Copyright
Copyright © 2018 by Mathematica Josephina, Inc.
Subject
Mathematics; Differential Geometry; Convex and Discrete Geometry; Fourier Analysis; Abstract Harmonic Analysis; Dynamical Systems and Ergodic Theory; Global Analysis and Analysis on Manifolds
ISSN
1050-6926
eISSN
1559-002X
D.O.I.
10.1007/s12220-018-0041-3
Publisher site
See Article on Publisher Site

Abstract

JGeomAnal https://doi.org/10.1007/s12220-018-0041-3 Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface 1 2 Mounir Nisse · Timur Sadykov Received: 17 November 2017 © Mathematica Josephina, Inc. 2018 Abstract Given a complex algebraic hypersurface H , we introduce a subset of the Newton polytope of the defining polynomial for H which is a polyhedral complex and enjoys the key topological and combinatorial properties of the amoeba of H for a large class of hypersurfaces. We provide an explicit formula for this polyhedral complex in the case when the spine of the amoeba is dual to a triangulation of the Newton polytope of the defining polynomial. In particular, this yields a description of the polyhedral complex when the hypersurface is optimal (Forsberg et al. in Adv Math 151:45–70, 2000). We conjecture that a polyhedral complex with these properties exists in general. Keywords Amoebas · Newton polytope · Tropical geometry · Polyhedral complex Mathematics Subject Classification 32A60 · 52B55 The presented research has been performed in the framework of the basic part of the scientific research state task in the field of scientific activity of the Ministry of Education and Science of the Russian Federation, Project No. 2.9577.2017/8.9. B Timur Sadykov Sadykov.TM@rea.ru Mounir Nisse

Journal

The Journal of Geometric AnalysisSpringer Journals

Published: Jun 4, 2018

References

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