Geometry of generalized amoebas

We show that the amoeba of a generic complex algebraic variety of codimension $1<r<n$ do not have a finite basis. In other words, it is not the intersection of finitely many hypersurface amoebas. Moreover we give a geometric characterization of the topological boundary of hypersurface amoebas refining an earlier result of F. Schroeter and T. de Wolff \cite{SW-13}.

An $n$-dimensional algebraic variety in $({\mathbb C}^\times)^{2n}$ covers its amoeba as well as its coamoeba generically finite-to-one. We provide an upper bound for the volume of these amoebas as well as for the number of points in the inverse images under the amoeba and coamoeba maps.

We present algorithms for computation and visualization of amoebas, their contours, compactified amoebas and sections of three-dimensional amoebas by two-dimensional planes. We also provide method and an algorithm for the computation of~polynomials whose amoebas exhibit the most complicated topology among all polynomials with a fixed Newton polytope. The presented algorithms are implemented in computer algebra systems Matlab 8 and Mathematica 9.

Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been intensively studied in recent years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of $n$ hypersurfaces in $(\mathbb{C}^*)^n$, which are canonical supersets of amoebas given by non-hypersurface varieties. Our main results...

An amoeba is the image of a subvariety of an algebraic torus under the logarithmic moment map. We consider some qualitative aspects of amoebas, establishing some results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semi-algebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba that is not of full dimension is not such a ...

The Ronkin function plays a fundamental role in the theory of amoebas. We introduce an analogue of the Ronkin function in the setting of coamoebas. It turns out to be closely related to a certain toric arrangement known as the shell of the coamoeba and we use our Ronkin type function to obtain some properties of it.

It is shown that tube sets over amoebas of algebraic varieties (and, more generally, of almost periodic holomorphic chains) of dimension q are q-pseudoconcave in the sense of Rothstein. This is a direct consequence of a representation of such sets as supports of positive closed currents.

We give a complete description of amoebas and coamoebas of $k$-dimensional very affine linear spaces in $(\mathbb{C}^*)^{n}$. This include an upper bound of their dimension, and we show that if a $k$-dimensional very affine linear space in $(\mathbb{C}^*)^{n}$ is generic, then the dimension of its (co)amoeba is equal to $\min \{ 2k, n\}$. Moreover, we prove that the volume of its coamoeba is equal to $\pi^{2k}$. In addition, if the space is generic and real, then the volume o...

The paper deals with singularities of nonconfluent hypergeometric functions in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of amoebas and the Newton polytopes of their defining polynomials. In particular, we show that all $\mathcal{A}$-discriminantal hypersurfaces (in the sense of Gelfand, Kapranov and Zelevinsky) have solid amoebas, that is, amoeba...

In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via tropical varieties. When we apply this construction to the Teichmuller spaces we see that they can be mapped in a real algebraic hypersurface in such a way that the cone over the boundary is a subpolyhedron of a tropical hypersurface. We wa...