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

: $\operatorname: \big( \setminus \\big)^n \to \mathbb R^n$

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
: $\operatorname(z_1, z_2, \dots, z_n) = \big(\log, z_1, , \log, z_2, , \dots, \log, z_n, \big).$

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

: $\mathcal A_p = \left\.$

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.Itenberg et al (2007) p. 3.
* 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

: $N_p(x) = \frac \int_ \log, p(z), \,\frac \wedge \frac \wedge \cdots \wedge \frac,$

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

: $N_p(x) = \frac \int_ \log, p(z), \,d\theta_1 \,d\theta_2 \cdots d\theta_n,$

where

: $z = \left(e^, e^, \dots, e^\right).$

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

: $p(z) = a z_1^ z_2^ \dots z_n^$

with $a \ne 0$ is

: $N_p(x) = \log, a, + k_1 x_1 + k_2 x_2 + \cdots + k_n x_n.$

References

*
* .

Further reading

*

External links

Amoebas of algebraic varieties

{{DEFAULTSORT:Amoeba (Mathematics)
Algebraic geometry