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## Summary

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, especially tropical geometry. The amoeba of P(zw) = w − 2z − 1 The amoeba of P(zw) = 3z2 + 5zw + w3 + 1. Notice the "vacuole" in the middle of the amoeba. The amoeba of P(zw) = 1 + z + z2 + z3 + z2w3 + 10zw + 12z2w + 10z2w2 The amoeba of P(zw) = 50z3 + 83z2w + 24zw2 + w3 + 392z2 + 414zw + 50w2 − 28z + 59w − 100 Points in the amoeba of P(xyz) = x + y + z − 1. Note that the amoeba is actually 3-dimensional, and not a surface (this is not entirely evident from the image).

## Definition

Consider the function

$\operatorname {Log} :{\big (}{\mathbb {C} }\setminus \{0\}{\big )}^{n}\to \mathbb {R} ^{n}$

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

$\operatorname {Log} (z_{1},z_{2},\dots ,z_{n})={\big (}\log |z_{1}|,\log |z_{2}|,\dots ,\log |z_{n}|{\big )}.$

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

${\mathcal {A}}_{p}=\left\{\operatorname {Log} (z):z\in {\big (}\mathbb {C} \setminus \{0\}{\big )}^{n},p(z)=0\right\}.$

Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky.

## Properties

• Any amoeba is a closed set.
• Any connected component of the complement $\mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}$  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 narrow 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

$N_{p}:\mathbb {R} ^{n}\to \mathbb {R}$

by the formula

$N_{p}(x)={\frac {1}{(2\pi i)^{n}}}\int _{\operatorname {Log} ^{-1}(x)}\log |p(z)|\,{\frac {dz_{1}}{z_{1}}}\wedge {\frac {dz_{2}}{z_{2}}}\wedge \cdots \wedge {\frac {dz_{n}}{z_{n}}},$

where $x$  denotes $x=(x_{1},x_{2},\dots ,x_{n}).$  Equivalently, $N_{p}$  is given by the integral

$N_{p}(x)={\frac {1}{(2\pi )^{n}}}\int _{[0,2\pi ]^{n}}\log |p(z)|\,d\theta _{1}\,d\theta _{2}\cdots d\theta _{n},$

where

$z=\left(e^{x_{1}+i\theta _{1}},e^{x_{2}+i\theta _{2}},\dots ,e^{x_{n}+i\theta _{n}}\right).$

The Ronkin function is convex and 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 $a\neq 0$  is

$N_{p}(x)=\log |a|+k_{1}x_{1}+k_{2}x_{2}+\cdots +k_{n}x_{n}.$