# Amoeba (mathematics)

For amoebas in set theory, see amoeba order.
The amoeba of
P(z,w)=w-2z-1.
The amoeba of
P(z, w)=3z2+5zw+w3+1.
Notice the "vacuole" in the middle of the amoeba.
The amoeba of
P(z, w) = 1 + z+z2 + z3 + z2w3 + 10zw + 12z2w +10z2w2.
The amoeba of
P(z, w)=50 z3 +83 z2 w+24 z w2 +w3+392 z2 +414 z w+50 w2 -28 z +59 w-100.
Points in the amoeba of
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).

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

${\displaystyle \mathrm {Log} :\left({\mathbb {C} }\backslash \{0\}\right)^{n}\to \mathbb {R} ^{n}}$

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

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

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

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

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

## Properties

• Any amoeba is a closed set.
• Any connected component of the complement ${\displaystyle \mathbb {R} ^{n}\backslash {\mathcal {A}}_{p}}$ 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.

## 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

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

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 it is 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

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

## References

1. ^ Gelfand, I. M.; Kapranov, M.M.; Zelevinsky, A.V. (1994). Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Boston, MA: Birkhäuser. ISBN 0-8176-3660-9. Zbl 0827.14036.
2. ^ Itenberg et al (2007) p.3
3. ^ Gross, Mark (2004). "Amoebas of complex curves and tropical curves". In Guest, Martin. UK-Japan winter school 2004—Geometry and analysis towards quantum theory. Lecture notes from the school, University of Durham, Durham, UK, January 6–9, 2004. Seminar on Mathematical Sciences 30. Yokohama: Keio University, Department of Mathematics. pp. 24–36. Zbl 1083.14061.