Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface
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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.
KeywordsAmoebas Newton polytope Tropical geometry Polyhedral complex
Mathematics Subject Classification32A60 52B55
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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