Amoeba (mathematics) explained

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

Consider the function

\operatorname{Log}: ( { C} \setminus \{ 0 \} ) ^{n} \to 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\operatorname{Log}(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 l A _{p} is defined as the image of the set of zeros of p under Log, sol A _{p} = \left\{\operatorname{Log}(z) : z \in ( C \setminus \{ 0 \} ) ^{n,} p(z) = 0 \right\}.

Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky.^[1]

Properties

Let

V \subset ( C ^{*} ) ^{n} be the zero locus of a polynomialf(z) = \sum _{j} a _{j} z ^{j}

where

A \subset Z ^{n} is finite, a _{j} \in C and z ^{j} = z ^{j _{1}} _{1} … z ^{j _{n}} _{n} if z = (z _{1} , ... ,z _{n} ) and j = (j _{1} , ... ,j _{n} ) . Let \Delta _{f} be the Newton polyhedron of f , i.e.,\Delta _{f} = ConvexHull \{j \in A \mid a _{j} \ne 0 \}.

Then

• Any amoeba is a closed set.

R ^{n} \setminus l A _{p} 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.
• The number of connected components of the complement

R ^{n} \setminus l{A} _{p} is not greater than \#(\Delta _{f} \cap Z ^{n} ) and not less than the number of vertices of \Delta _{f} .^[3]

• There is an injection from the set of connected components of complement

R ^{n} \setminus l{A} _{p} to \Delta _{f} \cap Z ^{n} . The vertices of \Delta _{f} are in the image under this injection. A connected component of complement R ^{n} \setminus l{A} _{p} is bounded if and only if its image is in the interior of \Delta _{f} .^[4]

• If

V \subset ( C ^{*} ) ^{2} , then the area of l{A} _{p} (V) is not greater than \pi ^{2} Area (\Delta _{f} ) .^[5]

Ronkin function

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} \to R

by the formula

N _{p(x)} = (1)/(( 2 \pi i) ^{n}) \int ^{-1} _{\operatorname{Log} (x)} log |p(z)| (dz _{1})/(z _{1}) \wedge (dz _{2})/(z _{2}) \wedge … \wedge (dz _{n})/(z _{n}) ,

where

x denotes x = (x _{1,} x _{2,} ... , x _{n).} Equivalently, N _{p} is given by the integralN _{p(x)} = (1)/(( 2 \pi) ^{n}) \int _{[ 0 , 2 \pi] ^{n}} log |p(z)| d\theta _{1} d\theta _{2} … d\theta _{n,}

where

z = \left(e ^{x _{1+i\theta} _{1}} , e ^{x _{2+i\theta} _{2}} , ... , e ^{x _{n+i\theta} _{n}} \right).

The Ronkin function is convex and affine on each connected component of the complement of the amoeba of

p(z) .^[6]

As an example, the Ronkin function of a monomial

p(z) = a z ^{k _{1}} _{1} z ^{k _{2}} _{2} ... z ^{k _{n}} _{n}

with

a \ne 0 isN _{p(x)} = log |a| + k _{1} x _{1} + k _{2} x _{2} + … + k _{n} x _{n.}

References

1. Book: Gelfand . I. M. . Israel Gelfand . M. M. . Kapranov . A. V. . Zelevinsky . Andrei Zelevinsky . Discriminants, resultants, and multidimensional determinants . Birkhäuser . Boston, MA . 1994 . 0827.14036 . 0-8176-3660-9 . Mathematics: Theory & Applications.
2. Itenberg et al (2007) p. 3.
3. Itenberg et al (2007) p. 3.
4. Itenberg et al (2007) p. 3.
5. Itenberg et al (2007) p. 3.
6. Book: Gross, Mark . Amoebas of complex curves and tropical curves . 1083.14061 . Guest . Martin . 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 . Yokohama . Keio University, Department of Mathematics . Seminar on Mathematical Sciences . 30 . 24–36 . 2004 .

• Book: Itenberg . Ilia . Mikhalkin . Grigory . Shustin . Eugenii . Tropical algebraic geometry . 1162.14300 . Oberwolfach Seminars . 35 . Basel . Birkhäuser . 978-3-7643-8309-1 . 2007 .
• .

Further reading

• Theobald . Thorsten . Computing amoebas . 1100.14048 . Exp. Math. . 11 . 4 . 513–526 . 2002 . 10.1080/10586458.2002.10504703 .

External links

• Amoebas of algebraic varieties