Amoeba (mathematics)

In the field of complex analysis, an amoeba is a set derived from the image of a complex polynomial under the logarithmic map. Specifically, let p(z₁, z₂, ..., z_n) be a complex polynomial in n complex variables, z₁, z₂, ..., z_n. The amoeba of p is the image of the set {(z₁, z₂, ..., z_n) ∈ (ℂ{0})ⁿ : p(z₁, z₂, ..., z_n) = 0} under the map Amoeba: (ℂ{0})ⁿ → ℝⁿ defined by

Amoeba(z₁, z₂, ..., z_n) = (log |z₁|, log |z₂|, ..., log |z_n|).

In simpler terms, the amoeba is the logarithm of the absolute values of the roots of a polynomial. The shape of an amoeba is often characterized by tentacle-like projections that extend to infinity. These tentacles correspond to the behavior of the polynomial at infinity. The complement of the amoeba is a finite number of connected, convex sets. The vertices of these convex sets (in the complement) are closely related to the Newton polytope of the polynomial p. The theory of amoebas is related to tropical geometry, where amoebas provide a link between algebraic geometry and piecewise-linear geometry. The term "amoeba" was coined by Israel Gelfand, Mikhail Kapranov, and Andrei Zelevinsky due to the shape's resemblance to biological amoebas.