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 ${\displaystyle z=(z_{1},z_{2},\dots ,z_{n})}$ of non-zero complex numbers with values in the Euclidean space ${\displaystyle \mathbb {R} ^{n},}$ given by the formula

Here, 'log' denotes the natural logarithm. If p(z) is a polynomial in ${\displaystyle n}$ complex variables, its amoeba ${\displaystyle {\mathcal {A}}_{p}}$ is defined as the image of the set of zeros of p under Log, so

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