# Amoeba (mathematics)

The amoeba of
P(z,w)=w-2z-1.
The amoeba of
P(z, w)=3z2+5zw+w3+1.
Notice the "vacuole" in the middle of the amoeba.
The amoeba of
P(z, w) = 1 + z+z2 + z3 + z2w3 + 10zw + 12z2w +10z2w2.
The amoeba of
P(z, w)=50 z3 +83 z2 w+24 z w2 +w3+392 z2 +414 z w+50 w2 -28 z +59 w-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. There is independently a concept of "amoeba order" in set theory.

## Definition

Consider the function

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

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 ${\mathcal A}_p$ is defined as the image of the set of zeros of p under Log, so

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

Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky.[1]

## Properties

• Any amoeba is a closed set.
• Any connected component of the complement $\mathbb R^n\backslash {\mathcal A}_p$ 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

$N_p(x)=\frac{1}{(2\pi i)^n}\int_{\mathrm{Log}^{-1}(x)}\log|p(z)| \,\frac{dz_1}{z_1} \wedge \frac{d z_2}{z_2}\wedge\cdots \wedge \frac{d z_n}{z_n},$

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

$N_p(x)=\frac{1}{(2\pi)^n}\int_{[0, 2\pi]^n}\log|p(z)| \,d\theta_1\,d\theta_2 \cdots d\theta_n,$

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 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 $a\ne 0$ is

$N_p(x) = \log|a|+k_1x_1+k_2x_2+\cdots+k_nx_n.\,$

## Set theory

In set theory, the amoeba order is the set of pairs $\langle P,\varepsilon\rangle$ where $P$ is an open subset of the Euclidean unit square $[0,1]\times[0,1]$ with Lebesgue measure $\mu(P) < \varepsilon$. We order the elements of the amoeba order by $\langle P,\varepsilon\rangle\le\langle Q,\varepsilon^*\rangle \iff P\supseteq Q \hbox{ and } \varepsilon\le\varepsilon^*$.[2]

## References

1. ^ Gelfand, I. M.; M.M. Kapranov, A.V. Zelevinsky (1994). Discriminants, resultants, and multidimensional determinants. Boston: Birkhäuser. ISBN 0-8176-3660-9.
2. ^ This definition is from Benedikt Löwe, What is ... An Amoeba (2)? [1].