http://en.wikipedia.org/wiki/Amoeba_(mathematics)
From Wikipedia, the free encyclopedia
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].
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
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
The amoeba of
[edit]