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. There is independently a concept of "amoeba order" in set theory.
Consider the function
- Any amoeba is a closed set.
- Any connected component of the complement 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
by the formula
where denotes Equivalently, is given by the integral
The Ronkin function is convex, and it is affine on each connected component of the complement of the amoeba of
As an example, the Ronkin function of a monomial
- Gelfand, I. M.; M.M. Kapranov, A.V. Zelevinsky (1994). Discriminants, resultants, and multidimensional determinants. Boston: Birkhäuser. ISBN 0-8176-3660-9.
- This definition is from Benedikt Löwe, What is ... An Amoeba (2)? .
- Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars 35. Basel: Birkhäuser. ISBN 978-3-7643-8309-1. Zbl 1162.14300.
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