Amoeba (mathematics)

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

The amoeba of Notice the "vacuole" in the middle of the amoeba.

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

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

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 mathematics, an amoeba is a set associated with a polynomial in one or more complex variables. Amoebas have applications in algebraic geometry.

1 Definition
2 Properties
3 Ronkin function
4 References
5 External links

[edit] Definition

Consider the function

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

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

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

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

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

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

[edit] 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 narrowing towards infinity.

[edit] 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

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

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

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

[edit] References

^ Gelfand, I. M.; M.M. Kapranov, A.V. Zelevinsky (1994). Discriminants, resultants, and multidimensional determinants. Boston: Birkhäuser. ISBN 0817636609.