Amoeba (mathematics)

right|thumb|The amoeba of">
$p(z,\; w)=w-2z-1.,$
[

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right|thumb|The_amoeba_of
$p(z,\; w)=3z^2,$$+5zw+w^3+1.,$
Notice the "vacuole" in the middle of the amoeba.]

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right|thumb|The amoeba of ">
$P(z,\; w)=1\; +\; z,$$+\; z^2\; +\; z^3\; +\; z^2w^3,$$+\; 10zw\; +\; 12z^2w,$$+\; 10z^2w^2.,$

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right|thumb|The amoeba of">
$P(z,\; w)=50\; z^3,$$+83\; z^2\; w+24\; z\; w^2,$$+w^3+392\; z^2,$$+414\; z\; w+50\; w^2,$$-28\; z\; +59\; w-100.,$ In complex analysis, a branch of mathematics, an amoeba is a set associated with a polynomial in one or more complex variables. Amoebas have applications in algebraic geometry. There is independently a concept of "amoeba order" in set theory.

Definition

Consider the function

:

defined on the set of all "n"-tuples $z=(z\_1,\; z\_2,\; dots,\; z\_n)$ of non-zero complex numbers with values in the Euclidean space $mathbb\; R^n,$ given by the formula:$mbox\{Log\}(z\_1,\; z\_2,\; dots,\; z\_n)=\; (log\; |z\_1|,\; log|z\_2|,\; dots,\; log\; |z\_n|).,$

Here, 'log' denotes the natural logarithm. If $p(z)$ is a polynomial in $n$ complex variables, its amoeba $\{mathcal\; A\}\_p$ is defined as the image of the set of zeros of "p" under Log, so

:

Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky [cite book
last = Gelfand
first = I. M.
coauthors = M.M. Kapranov, A.V. Zelevinsky
title = Discriminants, resultants, and multidimensional determinants
publisher = Boston: Birkhäuser
date = 1994
pages =
isbn = 0817636609] .

Properties

* Any amoeba is a closed set.
* Any connected component of the complement is convex.
* The area of an amoeba of a not identically zero polynomial in two complex variables is finite.
* A two-dimensional amoeba has a number of "tentacles" which are infinitely long and exponentially narrowing towards infinity.

Ronkin function

A useful tool in studying amoebas is the Ronkin function. For $p(z)$ a polynomial in $n$ complex variables, one defines the Ronkin function

: $N\_p:mathbb\; R^n\; o\; mathbb\; R$

by the formula

: $N\_p(x)=frac\{1\}\{(2pi\; i)^n\}int\_\{mbox\{Log\}^\{-1\}(x)\}log|p(z)|\; ,frac\{dz\_1\}\{z\_1\}\; wedge\; frac\{d\; z\_2\}\{z\_2\}wedgecdots\; wedge\; frac\{d\; z\_n\}\{z\_n\},$

where $x$ denotes $x=(x\_1,\; x\_2,\; dots,\; x\_n).$ Equivalently, $N\_p$ is given by the integral

: $N\_p(x)=frac\{1\}\{(2pi)^n\}int\_\{\; [0,\; 2pi]\; ^n\}log|p(z)|\; ,d\; heta\_1,d\; heta\_2\; cdots\; d\; heta\_n,$

where

: $z=left(e^\{x\_1+i\; heta\_1\},\; e^\{x\_2+i\; heta\_2\},\; dots,\; e^\{x\_n+i\; heta\_n\}\; ight).$

The Ronkin function is convex, and it is affine on each connected component of the complement of the amoeba of $p(z).$

As an example, the Ronkin function of a monomial

: $p(z)=az\_1^\{k\_1\}z\_2^\{k\_2\}dots\; z\_n^\{k\_n\},$

with $a\; e\; 0$ is

: $N\_p(x)\; =\; log|a|+k\_1x\_1+k\_2x\_2+cdots+k\_nx\_n.,$

et theory

In set theory, the amoeba order is the set of pairs $langle\; P,varepsilon\; angle$ where $P$ is an open subset of the Euclidean unit square $[0,1]\; imes\; [0,1]$ with Lebesgue measure . We order the elements of the amoeba order by $langle\; P,varepsilon\; anglelelangle\; Q,varepsilon^*\; angle\; iff\; Psupseteq\; Q\; hbox\{\; and\; \}\; varepsilonlevarepsilon^*$. [This definition is from Benedikt Löwe, "What is ... An Amoeba (2)?" [http://www.math.uni-bonn.de/people/loewe/Publ/amoebaR.ps] .]

References

External links

* [http://www.math.tamu.edu/~sottile/MSRI/viro.html WHAT IS an amoeba?]
* [http://www.dm.unipi.it/~bertrand/amoeb-geotrop/node1.html Amoebas of algebraic varieties]