Amoeba (mathematics)

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} \ne 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) = 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

• .

References

1. Itenberg et al (2007) p. 3.
2. Gross, Mark. (2004). "UK-Japan winter school 2004—Geometry and analysis towards quantum theory. Lecture notes from the school, University of Durham, Durham, UK, 6–9 January 2004". Keio University, Department of Mathematics.