# Amoeba (mathematics)

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.

## 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

Let $V\subset (\mathbb {C} ^{*})^{n}$ be the zero locus of a polynomial
    $f(z)=\sum _{j\in A}a_{j}z^{j}$
where $A\subset \mathbb {Z} ^{n}$ is finite, $a_{j}\in \mathbb {C}$ and $z^{j}=z_{1}^{j_{1}}\cdots z_{n}^{j_{n}}$ if $z=(z_{1},\dots ,z_{n})$ and $j=(j_{1},\dots ,j_{n})$. Let $\Delta _{f}$ be the Newton polyhedron of $f$, i.e.,
    $\Delta _{f}={\text{Convex Hull}}\{j\in A\mid a_{j}\neq 0\}.$
Then
* 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.
* The number of connected components of the complement $\mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}$ is not greater than $\#(\Delta _{f}\cap \mathbb {Z} ^{n})$ and not less than the number of vertices of $\Delta _{f}$.
* There is an injection from the set of connected components of complement $\mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}$ to $\Delta _{f}\cap \mathbb {Z} ^{n}$. The vertices of $\Delta _{f}$ are in the image under this injection. A connected component of complement $\mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}$ is bounded if and only if its image is in the interior of $\Delta _{f}$.
* If $V\subset (\mathbb {C} ^{*})^{2}$, then the area of ${\mathcal {A}}_{p}(V)$ is not greater than $\pi ^{2}{\text{Area}}(\Delta _{f})$.

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