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
The Journal of Geometric Analysis – Springer Journals
Published: Jun 4, 2018
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