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
L o g : ( C ∖ { 0 } ) n → R n defined on the set of all n-tuples z = ( z 1 , z 2 , … , z n ) of non-zero complex numbers with values in the Euclidean space R n , given by the formula
L o g ( z 1 , z 2 , … , z n ) = ( log | z 1 | , log | z 2 | , … , log | z n | ) . Here, 'log' denotes the natural logarithm. If p(z) is a polynomial in n complex variables, its amoeba A p is defined as the image of the set of zeros of p under Log, so
A p = { L o g ( z ) : z ∈ ( C ∖ { 0 } ) n , p ( z ) = 0 } . Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky.
Any amoeba is a closed set.Any connected component of the complement R n ∖ A p is convex.The area of an amoeba of a not identically zero polynomial in two complex variables is finite.A two-dimensional amoeba has a number of "tentacles" which are infinitely long and exponentially narrow towards infinity.A useful tool in studying amoebas is the Ronkin function. For p(z) a polynomial in n complex variables, one defines the Ronkin function
N p : R n → R by the formula
N p ( x ) = 1 ( 2 π i ) n ∫ L o g − 1 ( x ) log | p ( z ) | d z 1 z 1 ∧ d z 2 z 2 ∧ ⋯ ∧ d z n z n , where x denotes x = ( x 1 , x 2 , … , x n ) . Equivalently, N p is given by the integral
N p ( x ) = 1 ( 2 π ) n ∫ [ 0 , 2 π ] n log | p ( z ) | d θ 1 d θ 2 ⋯ d θ n , where
z = ( e x 1 + i θ 1 , e x 2 + i θ 2 , … , e x n + i θ n ) . The Ronkin function is convex, and it is affine on each connected component of the complement of the amoeba of p ( z ) .
As an example, the Ronkin function of a monomial
p ( z ) = a z 1 k 1 z 2 k 2 … z n k n with a ≠ 0 is
N p ( x ) = log | a | + k 1 x 1 + k 2 x 2 + ⋯ + k n x n . 
