Amoeba (mathematics)

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, especially tropical geometry.
Definition.
Consider the function
defined on the set of all "n"-tuples formula_2 of non-zero complex numbers with values in the Euclidean space formula_3 given by the formula
Here, log denotes the natural logarithm. If "p"("z") is a polynomial in formula_5 complex variables, its amoeba formula_6 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.
Ronkin function.
A useful tool in studying amoebas is the Ronkin function. For "p"("z"), a polynomial in "n" complex variables, one defines the Ronkin function
by the formula
where formula_11 denotes formula_12 Equivalently, formula_13 is given by the integral
where
The Ronkin function is convex and affine on each connected component of the complement of the amoeba of formula_16.
As an example, the Ronkin function of a monomial
with formula_18 is