Geometry of generalized amoebas

The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some traces of amoebas were appearing from time to time, even before the formal introduction, as auxiliary tools in several problems. After 1994 amoebas have been seen and studied in several areas of mathematics, from algebraic geometry and topology...

This paper is a report based on the results obtained during a three months internship at the University of Pittsburgh by the first author and under the mentorship of the second author. The notion of an amoeba of a subvariety in a torus $(\mathbb{C}^*)^n$ has been extended to subvarieties of the general linear group $GL_n(\mathbb{C})$ by the second author and Manon. In this paper, we show some basic properties of these matrix amoebas, e.g. any such amoeba is closed and the con...

A generalization of the amoeba and the Ronkin function of a plane algebraic curve for a pair of harmonic functions on an algebraic curve with punctures is proposed. Extremal properties of $M$-curves are proved and connected with the spectral theory of difference operators with positive coefficients.

The paper deals with amoebas of $k$-dimensional algebraic varieties in the algebraic complex torus of dimension $n\geq 2k$. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of $k$-dimensional algebraic variety in $(\mathbb{C}^*)^{n}$, with $n\geq 2k$, is finite.

The computation of amoebas has been a challenging open problem for the last dozen years. The most natural approach, namely to compute an amoeba via its boundary, has not been practical so far since only a superset of the boundary, the contour, is understood in theory and computable in practice. We define and characterize the extended boundary of an amoeba, which is sensitive to some degenerations that the topological boundary does not detect. Our description of the extended...

This article is a survey on the topic of polynomial amoebas. We review results of papers written on the topic with an emphasis on its computational aspects. Polynomial amoebas have numerous applications in various domains of mathematics and physics. Computation of the amoeba for a given polynomial and describing its properties is in general a problem of high complexity. We overview existing algorithms for computing and depicting amoebas and geometrical objects associated with...

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical varieties whose behavior is governed by algebraic geometry over the so-called tropical semifield. Geometric aspects of tropical algebraic geometry are the content o...

Answering a question by Nisse and Sottile, we derive a formula for the dimension of the amoeba of an irreducible algebraic variety.

We review results of papers written on the topic of polynomial amoebas with an emphasis on computational aspects of the topic. The polynomial amoebas have a lot of applications in various domains of science. Computation of the amoeba for a given polynomial and describing its properties is in general a problem of formidable complexity. We describe the main algorithms for computing and depicting the amoebas and geometrical objects associated with them, such as contours and spin...

In this note, we investigate the maximal number of intersection points of a line with the contour of hypersurface amoebas in $\mathbb{R}^n$. We define the latter number to be the $\mathbb{R}$-degree of the contour. We also investigate the $\mathbb{R}$-degree of related sets such as the boundary of amoebas and the amoeba of the real part of hypersurfaces defined over $\mathbb{R}$. For all these objects, we provide bounds for the respective $\mathbb{R}$-degrees.