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.

Consider the function defined on the set of all n-tuples

z

)

{\displaystyle z=(z_{1},z_{2},\dots ,z_{n})}

of non-zero complex numbers with values in the Euclidean space

,

given by the formula Here, log denotes the natural logarithm.

If p(z) is a polynomial in

complex variables, its amoeba

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.

be the zero locus of a polynomial where

is finite,

be the Newton polyhedron of

, i.e., Then 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

denotes

Equivalently,

is given by the integral where The Ronkin function is convex and affine on each connected component of the complement of the amoeba of

[3] As an example, the Ronkin function of a monomial with

The amoeba of
The amoeba of Notice the " vacuole " in the middle of the amoeba.
The amoeba of
The amoeba of
Points in the amoeba of Note that the amoeba is actually 3-dimensional, and not a surface (this is not entirely evident from the image).