Amoeba (mathematics)

Short description: Set associated with a complex-valued polynomial

The amoeba of $P(z,w)=w-2z-1.$

The amoeba of $P(z,w)=3z^2+5zw+w^3+1.$ Notice the "vacuole" in the middle of the amoeba.

The amoeba of $P(z,w)=1+z+z^2+z^3+z^2{w}^3+10zw+12z^{2}w+10z^2{w}^2.$

The amoeba of $P(z,w)=50z^3+83z^{2}w+24z{w}^2+w^3+392z^2+414zw+50w^2-28z+59w-100.$

Points in the amoeba of $P(x,y,z)=x+y+z-1.$ Note that the amoeba is actually 3-dimensional, and not a surface (this is not entirely evident from the image).

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

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

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

$\mathrm{Log}(z_1,z_2,\dots ,z_n)=(\log |z_1|,\log |z_2|,\dots ,\log |z_n|).$

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

${{𝒜}}_p=\{\mathrm{Log}(z):z\in (\mathbb{ℂ}\setminus \{0\}{)}^n,p(z)=0\}.$

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 $A\subset {\mathbb{\mathbb{Z}}}^n$ 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 ${\mathrm{\Delta }}_f$ be the Newton polyhedron of $f$, i.e.,

${\mathrm{\Delta }}_f=\text{Convex Hull}\{j\in A\mid a_j\ne 0\}.$

Then

• Any amoeba is a closed set.
• Any connected component of the complement ${\mathbb{ℝ}}^n\setminus {{𝒜}}_p$ 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 ${\mathbb{ℝ}}^n\setminus {{𝒜}}_p$ is not greater than $\mathrm{\#}({\mathrm{\Delta }}_f\cap {\mathbb{\mathbb{Z}}}^n)$ and not less than the number of vertices of ${\mathrm{\Delta }}_f$.
• There is an injection from the set of connected components of complement ${\mathbb{ℝ}}^n\setminus {{𝒜}}_p$ to ${\mathrm{\Delta }}_f\cap {\mathbb{\mathbb{Z}}}^n$. The vertices of ${\mathrm{\Delta }}_f$ are in the image under this injection. A connected component of complement ${\mathbb{ℝ}}^n\setminus {{𝒜}}_p$ is bounded if and only if its image is in the interior of ${\mathrm{\Delta }}_f$.
• If $V\subset ({\mathbb{ℂ}}^{*}{)}^2$, then the area of ${{𝒜}}_p(V)$ is not greater than ${\pi }^2\text{Area}({\mathrm{\Delta }}_f)$.

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

$N_p(x)=\frac{1}{(2\pi i{)}^n}\underset{{\mathrm{Log}}^{-1}(x)}{\int }\log |p(z)|\frac{d{z}_1}{z_1}\wedge \frac{d{z}_2}{z_2}\wedge \cdots \wedge \frac{d{z}_n}{z_n},$

where $x$ denotes $x=(x_1,x_2,\dots ,x_n).$ Equivalently, $N_p$ is given by the integral

$N_p(x)=\frac{1}{(2\pi {)}^n}\underset{[0,2\pi {]}^n}{\int }\log |p(z)|d{\theta }_{1}d{\theta }_2\cdots d{\theta }_n,$

where

$z=(e^{x_1+i{\theta }_1},e^{x_2+i{\theta }_2},\dots ,e^{x_n+i{\theta }_n}).$

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

$p(z)=a{z}_1^{k_1}{z}_2^{k_2}\dots z_n^{k_n}$

with $a\ne 0$ is

$N_p(x)=\log |a|+k_1{x}_1+k_2{x}_2+\cdots +k_n{x}_n.$

References

• Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars. 35. Basel: Birkhäuser. ISBN 978-3-7643-8309-1. 
• Viro, Oleg (2002), "What Is ... An Amoeba?", Notices of the American Mathematical Society 49 (8): 916–917, https://www.ams.org/notices/200208/what-is.pdf .

Further reading

• Theobald, Thorsten (2002). "Computing amoebas". Exp. Math. 11 (4): 513–526. doi:10.1080/10586458.2002.10504703. https://eudml.org/doc/52814. 

External links

• Amoebas of algebraic varieties