---
tag: C9WE6180A7
kind: definition
name: Amoeba
msc: [14T10, 32A60]
defines: PSK6VJQR9B
uses: [09FTAET5Z4, 07GNX8GCYP, 04DXAFN2ES, CYTTX8KHC8, 0JG9W3KA6B]
sources:
  - url: https://en.wikipedia.org/wiki/Amoeba_(mathematics)
---

## Definition

Let $n \ge 1$ and let $\operatorname{Log} \colon (\C^\times)^n \to \R^n$ be the map
$$\operatorname{Log}(z_1, \dots, z_n) = (\log \abs{z_1}, \dots, \log \abs{z_n}).$$
Let $f$ be a nonzero Laurent polynomial in $z_1, \dots, z_n$ with [[complex|09FTAET5Z4]] coefficients, and let $V_f = \set{z \in (\C^\times)^n : f(z) = 0}$.
The *amoeba* of $f$ is the set $\mathcal{A}_f = \operatorname{Log}(V_f) \subseteq \R^n$.

## Remarks

The amoeba is a closed subset of $\R^n$, and each [[connected component|0JG9W3KA6B]] of its complement is convex.
Let $f$ be nonconstant.
The number of these components is at least the number of vertices of the [[Newton polytope|CYTTX8KHC8]] of $f$, and at most its number of lattice points.

Example: let $f = 1 + z + w$.
The point $(0, 0)$ lies in $\mathcal{A}_f$: the point $z = e^{2\pi i/3}$, $w = e^{-2\pi i/3}$ has $\abs{z} = \abs{w} = 1$ and $1 + z + w = 0$.

Non-example: the point $(1, 0)$ does not lie in $\mathcal{A}_f$.
It would need $\abs{z} = e$, $\abs{w} = 1$ and $z = -1 - w$, but $\abs{1 + w} \le 2 < e$.

In one variable, $f(z) = z - 2$ has $V_f = \set{2}$ and $\mathcal{A}_f = \set{\log 2}$.