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 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 n 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.

Properties

Let be the zero locus of a polynomial where is finite, and if and. Let \Delta_{f} be the Newton polyhedron of f, i.e., Then

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 x denotes Equivalently, N_p is given by the integral where The Ronkin function is convex and affine on each connected component of the complement of the amoeba of p(z). As an example, the Ronkin function of a monomial with a \ne 0 is