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.

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

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

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

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.^[1]

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}\neq 0\}.

Then

Any amoeba is a closed set.
Any connected component of the complement $\mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}$ 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 $\mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}$ is not greater than $\#(\Delta _{f}\cap \mathbb {Z} ^{n})$ and not less than the number of vertices of $\Delta _{f}$ .^[2]
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 $\Delta _{f}$ .^[2]
If $V\subset (\mathbb {C} ^{*})^{2}$ , then the area of ${\mathcal {A}}_{p}(V)$ is not greater than $\pi ^{2}{\text{Area}}(\Delta _{f})$ .^[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

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 $p(z)$ .^[3]

As an example, the Ronkin function of a monomial

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

with $a\neq 0$ is

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

References

^ 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.
^ ^a ^b ^c ^d Itenberg et al (2007) p. 3.
^ Gross, Mark (2004). "Amoebas of complex curves and tropical curves". In Guest, Martin (ed.). UK-Japan winter school 2004—Geometry and analysis towards quantum theory. Lecture notes from the school, University of Durham, Durham, UK, 6–9 January 2004. Seminar on Mathematical Sciences. Vol. 30. Yokohama: Keio University, Department of Mathematics. pp. 24–36. Zbl 1083.14061.

Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars. Vol. 35. Basel: Birkhäuser. ISBN 978-3-7643-8309-1. Zbl 1162.14300.
Viro, Oleg (2002), "What Is ... An Amoeba?" (PDF), Notices of the American Mathematical Society, 49 (8): 916–917.

External links

Amoebas of algebraic varieties

[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.

[IMS3-2] Itenberg et al (2007) p. 3.

[3] Gross, Mark (2004). "Amoebas of complex curves and tropical curves". In Guest, Martin (ed.). UK-Japan winter school 2004—Geometry and analysis towards quantum theory. Lecture notes from the school, University of Durham, Durham, UK, 6–9 January 2004. Seminar on Mathematical Sciences. Vol. 30. Yokohama: Keio University, Department of Mathematics. pp. 24–36. Zbl 1083.14061.

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