Timur Sadykov: Compactified and polyhedral versions of the amoeba of an algebraic hypersurface

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 algebraic variety.

Despite the simple definition, efficient computation of the amoeba of a given algebraic variety represents a task of formidable computational complexity. Various approaches have been recently tried to compute the shape of an amoeba or approximate it by simpler geometric objects. The fundamental problems addressed in numerous papers are the detection of the topological type of an amoeba, the membership problem for a given connected component of an amoeba complement, and detection of the order of such a component in the case of hypersurface amoebas.

Alongside with the definition of unbounded affine amoeba of an algebraic hypersurface, a competing definition of compactified amoeba has been introduced. While the affine amoeba of a hypersurface is its Reinhardt diagram in the logarithmic scale, its compactified amoeba is defined to be 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 (based on a joint work with Mounir Nisse) I will introduce 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.