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