Amoeba (mathematics)

Summary

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.

The amoeba of P(zw) = w − 2z − 1
The amoeba of P(zw) = 3z2 + 5zw + w3 + 1. Notice the "vacuole" in the middle of the amoeba.
The amoeba of P(zw) = 1 + z + z2 + z3 + z2w3 + 10zw + 12z2w + 10z2w2
The amoeba of P(zw) = 50z3 + 83z2w + 24zw2 + w3 + 392z2 + 414zw + 50w2 − 28z + 59w − 100
Points in the amoeba of P(xyz) = x + y + z − 1. Note that the amoeba is actually 3-dimensional, and not a surface (this is not entirely evident from the image).

DefinitionEdit

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]

PropertiesEdit

  • 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 functionEdit

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 affine on each connected component of the complement of the amoeba of  .[3]

As an example, the Ronkin function of a monomial

 

with   is

 

ReferencesEdit

  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 (ed.). UK-Japan winter school 2004—Geometry and analysis towards quantum theory. Lecture notes from the school, University of Durham, Durham, UK, 6–9 January 2004. Seminar on Mathematical Sciences. Vol. 30. Yokohama: Keio University, Department of Mathematics. pp. 24–36. Zbl 1083.14061.
  • Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars. Vol. 35. Basel: Birkhäuser. ISBN 978-3-7643-8309-1. Zbl 1162.14300.
  • Viro, Oleg (2002), "What Is ... An Amoeba?" (PDF), Notices of the American Mathematical Society, 49 (8): 916–917.

Further readingEdit

  • Theobald, Thorsten (2002). "Computing amoebas". Exp. Math. 11 (4): 513–526. doi:10.1080/10586458.2002.10504703. Zbl 1100.14048.

External linksEdit

  • Amoebas of algebraic varieties