Amoeba (mathematics)

For amoebas in set theory, see amoeba order.

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.

The amoeba of ${\displaystyle P(z,w)=w-2z-1.}$

The amoeba of ${\displaystyle P(z,w)=3z^{2}+5zw+w^{3}+1.}$ Notice the "vacuole" in the middle of the amoeba.

The amoeba of ${\displaystyle P(z,w)=1+z+z^{2}+z^{3}+z^{2}w^{3}+10zw+12z^{2}w+10z^{2}w^{2}.}$

The amoeba of ${\displaystyle P(z,w)=50z^{3}+83z^{2}w+24zw^{2}+w^{3}+392z^{2}+414zw+50w^{2}-28z+59w-100.}$

Points in the amoeba of ${\displaystyle P(x,y,z)=x+y+z-1.}$ Note that the amoeba is actually 3-dimensional, and not a surface (this is not entirely evident from the image).

Definition

Consider the function

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

defined on the set of all n-tuples ${\displaystyle z=(z_{1},z_{2},\dots ,z_{n})}$ of non-zero complex numbers with values in the Euclidean space ${\displaystyle \mathbb {R} ^{n},}$ given by the formula

${\displaystyle \operatorname {Log} (z_{1},z_{2},\dots ,z_{n})={\big (}\log |z_{1}|,\log |z_{2}|,\dots ,\log |z_{n}|{\big )}.}$

Here, log denotes the natural logarithm. If p(z) is a polynomial in ${\displaystyle n}$ complex variables, its amoeba ${\displaystyle {\mathcal {A}}_{p}}$ is defined as the image of the set of zeros of p under Log, so

${\displaystyle {\mathcal {A}}_{p}=\left\{\operatorname {Log} (z):z\in {\big (}\mathbb {C} \setminus \{0\}{\big )}^{n},p(z)=0\right\}.}$

Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky.^[1]

Properties

Let ${\displaystyle V\subset (\mathbb {C} ^{*})^{n}}$ be the zero locus of a polynomial

${\displaystyle f(z)=\sum _{j\in A}a_{j}z^{j}}$

where ${\displaystyle A\subset \mathbb {Z} ^{n}}$ is finite, ${\displaystyle a_{j}\in \mathbb {C} }$ and ${\displaystyle z^{j}=z_{1}^{j_{1}}\cdots z_{n}^{j_{n}}}$ if ${\displaystyle z=(z_{1},\dots ,z_{n})}$ and ${\displaystyle j=(j_{1},\dots ,j_{n})}$. Let ${\displaystyle \Delta _{f}}$ be the Newton polyhedron of ${\displaystyle f}$, i.e.,

${\displaystyle \Delta _{f}={\text{Convex Hull}}\{j\in A\mid a_{j}\neq 0\}.}$

Then

• Any amoeba is a closed set.
• Any connected component of the complement ${\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}}$ is convex.^[2]
• 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 ${\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}}$ is not greater than ${\displaystyle \#(\Delta _{f}\cap \mathbb {Z} ^{n})}$ and not less than the number of vertices of ${\displaystyle \Delta _{f}}$.^[2]
• There is an injection from the set of connected components of complement ${\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}}$ to ${\displaystyle \Delta _{f}\cap \mathbb {Z} ^{n}}$. The vertices of ${\displaystyle \Delta _{f}}$ are in the image under this injection. A connected component of complement ${\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}}$ is bounded if and only if its image is in the interior of ${\displaystyle \Delta _{f}}$.^[2]
• If ${\displaystyle V\subset (\mathbb {C} ^{*})^{2}}$, then the area of ${\displaystyle {\mathcal {A}}_{p}(V)}$ is not greater than ${\displaystyle \pi ^{2}{\text{Area}}(\Delta _{f})}$.^[2]

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

${\displaystyle N_{p}:\mathbb {R} ^{n}\to \mathbb {R} }$

by the formula

${\displaystyle N_{p}(x)={\frac {1}{(2\pi i)^{n}}}\int _{\operatorname {Log} ^{-1}(x)}\log |p(z)|\,{\frac {dz_{1}}{z_{1}}}\wedge {\frac {dz_{2}}{z_{2}}}\wedge \cdots \wedge {\frac {dz_{n}}{z_{n}}},}$

where ${\displaystyle x}$ denotes ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n}).}$ Equivalently, ${\displaystyle N_{p}}$ is given by the integral

${\displaystyle N_{p}(x)={\frac {1}{(2\pi )^{n}}}\int _{[0,2\pi ]^{n}}\log |p(z)|\,d\theta _{1}\,d\theta _{2}\cdots d\theta _{n},}$

where

${\displaystyle z=\left(e^{x_{1}+i\theta _{1}},e^{x_{2}+i\theta _{2}},\dots ,e^{x_{n}+i\theta _{n}}\right).}$

The Ronkin function is convex and affine on each connected component of the complement of the amoeba of ${\displaystyle p(z)}$.^[3]

As an example, the Ronkin function of a monomial

${\displaystyle p(z)=az_{1}^{k_{1}}z_{2}^{k_{2}}\dots z_{n}^{k_{n}}}$

with ${\displaystyle a\neq 0}$ is

${\displaystyle N_{p}(x)=\log |a|+k_{1}x_{1}+k_{2}x_{2}+\cdots +k_{n}x_{n}.}$