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


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

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


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