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Amoeba (mathematics) The amoeba of  Notice the "vacuole" in the middle of the amoeba. The amoeba of     The amoeba of     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.

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Definition

Consider the function 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 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.

Properties

• Any 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 function

A 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 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 is Set theory

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

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)? .