Amoeba (mathematics)

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For amoebas in set theory, see amoeba order.
The amoeba of
P(z,w)=w-2z-1.
The amoeba of
P(z, w)=3z2+5zw+w3+1.
Notice the "vacuole" in the middle of the amoeba.
The amoeba of
P(z, w) = 1 + z+z2 + z3 + z2w3 + 10zw + 12z2w +10z2w2.
The amoeba of
P(z, w)=50 z3 +83 z2 w+24 z w2 +w3+392 z2 +414 z w+50 w2 -28 z +59 w-100.
Points in the amoeba of
P(x,y,z)=x+y+z-1. Note that the amoeba is actually 3-dimensional, and not a surface, (this is not entirely evident from the image).

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[edit]

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 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.[1]

Properties[edit]

  • Any amoeba is a closed set.
  • Any connected component of the complement is convex.[2]
  • 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.

Ronkin function[edit]

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

where

The Ronkin function is convex, and it is affine on each connected component of the complement of the amoeba of .[3]

As an example, the Ronkin function of a monomial

with is

References[edit]

  1. ^ Gelfand, I. M.; Kapranov, M.M.; Zelevinsky, A.V. (1994). Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Boston, MA: Birkhäuser. ISBN 0-8176-3660-9. Zbl 0827.14036. 
  2. ^ Itenberg et al (2007) p.3
  3. ^ Gross, Mark (2004). "Amoebas of complex curves and tropical curves". In Guest, Martin. UK-Japan winter school 2004—Geometry and analysis towards quantum theory. Lecture notes from the school, University of Durham, Durham, UK, January 6–9, 2004. Seminar on Mathematical Sciences 30. Yokohama: Keio University, Department of Mathematics. pp. 24–36. Zbl 1083.14061. 

Further reading[edit]

External links[edit]