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Amoeba (mathematics)

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In complex analysis, a branch of mathematics, an amoeba is a set associated with apolynomial in one or more complex variables. Amoebas have applications in algebraic geometry. There is independently a concept of "amoeba order" in set theory.

p(z, w)=w-2z-1.\,

The amoeba of

DefinitionConsider the function

\mbox{Log}: \left({\mathbb C}\backslash\{0\}\right)^n \to \mathbb R^n
z=(z_1, z_2, \dots, z_n)
\mathbb R^n,

defined on the set of all n-tuples of non-zero complex numberswith values in the Euclidean space given by the formula

Here, 'log' denotes the natural logarithm. If p(z) is a polynomial in n complex variables, itsamoeba

\mbox{Log}(z_1, z_2, \dots, z_n)= (\log |z_1|, \log|z_2|, \dots, \log |z_n|).\,
{\mathcal A}_p

is defined as the image of the set of zeros of p under Log, so

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

    • PropertiesAny amoeba is a closed set.
    • Any connected component of the complement 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 narrowing towards infinity.

Ronkin functionA useful tool in studying amoebas is the Ronkin function. For p(z) a polynomial in ncomplex variables, one defines the Ronkin function

by the formula

where x denotes

{\mathcal A}_p = \left\{\mbox{Log} (z) \, : \, z\in \left({\mathbb C}\backslash\{0\}\right)^n, p(z)=0\right\}.\,
N_p:\mathbb R^n \to \mathbb R
N_p(x)=\frac{1}{(2\pi i)^n}\int_{\mbox{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},
\mathbb R^n\backslash {\mathcal A}_p

The amoeba of Notice the "vacuole" in the middle of the amoeba.

p(z, w)=3z^2\,
+5zw+w^3+1.\,
P(z, w)=1 + z\,
+ z^2 + z^3 + z^2w^3\,

Equivalently, Np is given by the integral

where

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

with

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,
z=\left(e^{x_1+i\theta_1}, e^{x_2+i\theta_2}, \dots, e^{x_n+i\theta_n}\right).
p(z)=az_1^{k_1}z_2^{k_2}\dots z_n^{k_n}\,
x=(x_1, x_2, \dots, x_n).

The amoeba of

+ 10zw + 12z^2w\,
+ 10z^2w^2.\,
a\ne 0

is

Set theory

where P is an open subset of the Euclidean unit square with Lebesgue measure . We order the elements of the amoeba order by

In set theory, the amoeba order is the set of pairs

.[2]

N_p(x) = \log|a|+k_1x_1+k_2x_2+\cdots+k_nx_n.\,
\langle P,\varepsilon\rangle\le\langle Q,\varepsilon^*\rangle \iff P\supseteq Q \hbox{ and } \varepsilon\le\varepsilon^*
\langle P,\varepsilon\rangle
[0,1]\times[0,1]
\mu(P) < \varepsilon
P(z, w)=50 z^3\,
+83 z^2 w+24 z w^2\,
+w^3+392 z^2\,
+414 z w+50 w^2\,
-28 z +59 w-100.\,

The amoeba of

References

    1. ^ Gelfand, I. M.; M.M. Kapranov, A.V. Zelevinsky (1994). Discriminants, resultants, and multidimensional determinants. Boston: Birkhäuser. ISBN 0817636609.
    2. ^ This definition is from Benedikt Löwe, What is ... An Amoeba (2)? [1].

External links

Categories: Algebraic geometry