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The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1356–1368 | Cite as

Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface

  • Mounir Nisse
  • Timur SadykovEmail author
Article
  • 29 Downloads

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 

Notes

Acknowledgements

A large part of this paper was written during Timur Sadykov’s visits to Seoul in 2017. The authors thank Korea Institute for Advanced Study for providing excellent conditions for research and writing.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.School of MathematicsKorea Institue for Advanced StudySeoulRepublic of Korea
  2. 2.Department of Mathematics and Computer SciencePlekhanov Russian UniversityMoscowRussia

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