Abstract:
Amoebas of complex algebraic varieties have attracted substantial attention in the recent years after their inception in the work of Gelfand, Kapranov and Zelevinsky. Being a semi-analytic subset of the real space, the amoeba carries a lot of geometric, algebraic, topological, and combinatorial information on the corresponding (tropical) algebraic variety.
Alongside with the definition of unbounded affine amoeba of an algebraic hypersurface, an alternative definition of compactified amoeba has been introduced.
While the affine amoeba of a hypersurface is defined to be its Reinhardt diagram in the logarithmic scale, the compactified amoeba is the image of the hypersurface under the moment map providing a homeomorphism between the Newton polytope of the defining polynomial of that hypersurface and the positive orthant of the real vector space. Being topologically equivalent to the standard affine amoeba, its compactified counterpart often has the substantial disadvantage of exhibiting complement components of very different relative size. This makes it difficult to work with compactified amoebas in a computationally reliable way and probably explains the focus of research on affine amoebas.
In the talk (mainly based on a joint work with Dmitry Bogdanov) I will discuss the definition of an amoeba-shaped polyhedral complex of an algebraic hypersurface. Like the compactified amoeba, this polyhedral complex is a subset of the Newton polytope of the defining polynomial of the hypersurface. Besides, it is a deformation retract of the compactified amoeba and provides the straightforward solution to the membership problem: the order of a connected component in its complement is itself a point in this component. An explicit formula for this polyhedral complex will be given in the case when the hypersurface is optimal. Furthermore, the tropical-geometric obstructions for the amoeba of a multivariate hypergeometric polynomial to be optimal will be introduced and discussed.
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