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

\mathrm{Log}: \left({\mathbb C}\backslash\{0\}\right)^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

\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 $n$ 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\{\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 $\mathbb R^n\backslash {\mathcal A}_p$ 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

N_p:\mathbb R^n \to \mathbb R

by the formula

N_p(x)=\frac{1}{(2\pi i)^n}\int_{\mathrm{Log}^{-1}(x)}\log|p(z)| \,\frac{dz_1}{z_1} \wedge \frac{d z_2}{z_2}\wedge\cdots \wedge \frac{d z_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 it is 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 $a\ne 0$ is

N_p(x) = \log|a|+k_1x_1+k_2x_2+\cdots+k_nx_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.
↑ Itenberg et al (2007) p.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.

Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars 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

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Amoebas of algebraic varieties

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