Amoeba

Remarks

The amoeba is a closed subset of $\R^n$, and each connected component 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 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 $\mathcal{A}_f = \set{\log 2}$.