Amoeba (mathematics)

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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. There is independently a concept of "amoeba order" in set theory.

Definition[edit]

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[edit]

  • Any amoeba is a closed set.
  • Any connected component of the complement \mathbb R^n\backslash {\mathcal A}_p is convex.
  • 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[edit]

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).

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.\,

Set theory[edit]

In set theory, the amoeba order is the set of pairs \langle P,\varepsilon\rangle where P is an open subset of the Euclidean unit square [0,1]\times[0,1] with Lebesgue measure \mu(P) < \varepsilon. We order the elements of the amoeba order by \langle P,\varepsilon\rangle\le\langle Q,\varepsilon^*\rangle \iff P\supseteq Q \hbox{ and } \varepsilon\le\varepsilon^*.[2]

References[edit]

  1. ^ Gelfand, I. M.; M.M. Kapranov, A.V. Zelevinsky (1994). Discriminants, resultants, and multidimensional determinants. Boston: Birkhäuser. ISBN 0-8176-3660-9. 
  2. ^ This definition is from Benedikt Löwe, What is ... An Amoeba (2)? [1].

Further reading[edit]

  • Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars 35. Basel: Birkhäuser. ISBN 978-3-7643-8309-1. Zbl 1162.14300. 

External links[edit]