ID: math/0108225

Amoebas of algebraic varieties

August 31, 2001

View on ArXiv

Similar papers 2

Amoebas of half-dimensional varieties

December 15, 2014

85% Match
Grigory Mikhalkin
Algebraic Geometry

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.

Find SimilarView on arXiv

Non-archimedean amoebas and tropical varieties

August 23, 2004

85% Match
Manfred Einsiedler, Mikhail Kapranov, Douglas Lind
Algebraic Geometry
Commutative Algebra
Dynamical Systems

We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and estab...

Find SimilarView on arXiv

Approximating amoebas and coamoebas by sums of squares

January 21, 2011

85% Match
Thorsten Theobald, Wolff Timo de
Algebraic Geometry

Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming. Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic f...

Find SimilarView on arXiv

Geometric and Combinatorial Structure of Hypersurface Coamoebas

June 15, 2009

85% Match
Mounir Nisse
Algebraic Geometry
Complex Variables
Geometric Topology

Let $V$ be a complex algebraic hypersurface defined by a polynomial $f$ with Newton polytope $\Delta$. It is well known that the spine of its amoeba has a structure of a tropical hypersurface. We prove in this paper that there exists a complex tropical hypersurface $V_{\infty, f}$ such that its coamoeba is homeomorphic to the closure in the real torus of the coamoeba of $V$. Moreover, the coamoeba of $V_{\infty, f}$ contains an arrangement of $(n-1)$-torus depending only on t...

Find SimilarView on arXiv

Describing Amoebas

May 1, 2018

85% Match
Mounir Nisse, Frank Sottile
Algebraic Geometry

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 ...

Find SimilarView on arXiv

A Nullstellensatz for amoebas

March 8, 2006

85% Match
Kevin Purbhoo
Algebraic Geometry
Complex Variables

The amoeba of an affine algebraic variety V in (C^*)^r is the image of V under the map (z_1, ..., z_r) -> (log|z_1|, ..., log|z_r|). We give a characterisation of the amoeba based on the triangle inequality, which we call testing for lopsidedness. We show that if a point is outside the amoeba of V, there is an element of the defining ideal which witnesses this fact by being lopsided. This condition is necessary and sufficient for amoebas of arbitrary codimension, as well as f...

Find SimilarView on arXiv

Amoeba-shaped polyhedral complex of an algebraic hypersurface

August 23, 2017

84% Match
Mounir Nisse, Timur Sadykov
Algebraic Geometry

Given a complex algebraic hypersurface~$H$, we introduce a polyhedral complex which is a subset of the Newton polytope of the defining polynomial for~$H$ and enjoys the key topological and combinatorial properties of the amoeba of~$H.$ 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...

Find SimilarView on arXiv

Amoeba finite basis does not exist in general

March 16, 2014

84% Match
Mounir Nisse
Algebraic Geometry
Complex Variables

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}.

Find SimilarView on arXiv

A remark on amoebas in higher codimensions

January 3, 2008

84% Match
Alexander Rashkovskii
Complex Variables

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.

Find SimilarView on arXiv

Intersections of Amoebas

October 28, 2015

84% Match
Martina Juhnke-Kubitzke, Wolff Timo de
Combinatorics
Algebraic Geometry

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...

Find SimilarView on arXiv