The amoeba of P ( z , w ) = w − 2 z − 1. {\displaystyle P(z,w)=w-2z-1.}
The amoeba of P ( z , w ) = 3 z 2 + 5 z w + w 3 + 1. {\displaystyle P(z,w)=3z^{2}+5zw+w^{3}+1.} Notice the "vacuole" in the middle of the amoeba.
The amoeba of P ( z , w ) = 1 + z + z 2 + z 3 + z 2 w 3 + 10 z w + 12 z 2 w + 10 z 2 w 2 . {\displaystyle P(z,w)=1+z+z^{2}+z^{3}+z^{2}w^{3}+10zw+12z^{2}w+10z^{2}w^{2}.}
The amoeba of P ( z , w ) = 50 z 3 + 83 z 2 w + 24 z w 2 + w 3 + 392 z 2 + 414 z w + 50 w 2 − 28 z + 59 w − 100. {\displaystyle P(z,w)=50z^{3}+83z^{2}w+24zw^{2}+w^{3}+392z^{2}+414zw+50w^{2}-28z+59w-100.}
Points in the amoeba of P ( x , y , z ) = x + y + z − 1. {\displaystyle 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

Consider the function

Log : ( C ∖ { 0 } ) n → R n {\displaystyle \operatorname {Log} :{\big (}{\mathbb {C} }\setminus \{0\}{\big )}^{n}\to \mathbb {R} ^{n}}

defined on the set of all n-tuples z = ( z 1 , z 2 , … , z n ) {\displaystyle z=(z_{1},z_{2},\dots ,z_{n})} of non-zero complex numbers with values in the Euclidean space R n , {\displaystyle \mathbb {R} ^{n},} given by the formula

Log ⁡ ( z 1 , z 2 , … , z n ) = ( log ⁡ | z 1 | , log ⁡ | z 2 | , … , log ⁡ | z n | ) . {\displaystyle \operatorname {Log} (z_{1},z_{2},\dots ,z_{n})={\big (}\log |z_{1}|,\log |z_{2}|,\dots ,\log |z_{n}|{\big )}.}

Here, log denotes the natural logarithm. If p(z) is a polynomial in n {\displaystyle n} complex variables, its amoeba A p {\displaystyle {\mathcal {A}}_{p}} is defined as the image of the set of zeros of p under Log, so

A p = { Log ⁡ ( z ) : z ∈ ( C ∖ { 0 } ) n , p ( z ) = 0 } . {\displaystyle {\mathcal {A}}_{p}=\left\{\operatorname {Log} (z):z\in {\big (}\mathbb {C} \setminus \{0\}{\big )}^{n},p(z)=0\right\}.}

Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky.

Properties

Let V ⊂ ( C ∗ ) n {\displaystyle V\subset (\mathbb {C} ^{*})^{n}} be the zero locus of a polynomial

f ( z ) = ∑ j ∈ A a j z j {\displaystyle f(z)=\sum _{j\in A}a_{j}z^{j}}

where A ⊂ Z n {\displaystyle A\subset \mathbb {Z} ^{n}} is finite, a j ∈ C {\displaystyle a_{j}\in \mathbb {C} } and z j = z 1 j 1 ⋯ z n j n {\displaystyle z^{j}=z_{1}^{j_{1}}\cdots z_{n}^{j_{n}}} if z = ( z 1 , … , z n ) {\displaystyle z=(z_{1},\dots ,z_{n})} and j = ( j 1 , … , j n ) {\displaystyle j=(j_{1},\dots ,j_{n})}. Let Δ f {\displaystyle \Delta _{f}} be the Newton polyhedron of f {\displaystyle f}, i.e.,

Δ f = Convex Hull { j ∈ A ∣ a j ≠ 0 } . {\displaystyle \Delta _{f}={\text{Convex Hull}}\{j\in A\mid a_{j}\neq 0\}.}

Then

  • Any amoeba is a closed set.
  • Any connected component of the complement R n ∖ A p {\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {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.
  • The number of connected components of the complement R n ∖ A p {\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}} is not greater than # ( Δ f ∩ Z n ) {\displaystyle \#(\Delta _{f}\cap \mathbb {Z} ^{n})} and not less than the number of vertices of Δ f {\displaystyle \Delta _{f}}.
  • There is an injection from the set of connected components of complement R n ∖ A p {\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}} to Δ f ∩ Z n {\displaystyle \Delta _{f}\cap \mathbb {Z} ^{n}}. The vertices of Δ f {\displaystyle \Delta _{f}} are in the image under this injection. A connected component of complement R n ∖ A p {\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}} is bounded if and only if its image is in the interior of Δ f {\displaystyle \Delta _{f}}.
  • If V ⊂ ( C ∗ ) 2 {\displaystyle V\subset (\mathbb {C} ^{*})^{2}}, then the area of A p ( V ) {\displaystyle {\mathcal {A}}_{p}(V)} is not greater than π 2 Area ( Δ f ) {\displaystyle \pi ^{2}{\text{Area}}(\Delta _{f})}.

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 → R {\displaystyle N_{p}:\mathbb {R} ^{n}\to \mathbb {R} }

by the formula

N p ( x ) = 1 ( 2 π i ) n ∫ Log − 1 ⁡ ( x ) log ⁡ | p ( z ) | d z 1 z 1 ∧ d z 2 z 2 ∧ ⋯ ∧ d z n z n , {\displaystyle N_{p}(x)={\frac {1}{(2\pi i)^{n}}}\int _{\operatorname {Log} ^{-1}(x)}\log |p(z)|\,{\frac {dz_{1}}{z_{1}}}\wedge {\frac {dz_{2}}{z_{2}}}\wedge \cdots \wedge {\frac {dz_{n}}{z_{n}}},}

where x {\displaystyle x} denotes x = ( x 1 , x 2 , … , x n ) . {\displaystyle x=(x_{1},x_{2},\dots ,x_{n}).} Equivalently, N p {\displaystyle 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 , {\displaystyle N_{p}(x)={\frac {1}{(2\pi )^{n}}}\int _{[0,2\pi ]^{n}}\log |p(z)|\,d\theta _{1}\,d\theta _{2}\cdots d\theta _{n},}

where

z = ( e x 1 + i θ 1 , e x 2 + i θ 2 , … , e x n + i θ n ) . {\displaystyle z=\left(e^{x_{1}+i\theta _{1}},e^{x_{2}+i\theta _{2}},\dots ,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 ) {\displaystyle 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 {\displaystyle p(z)=az_{1}^{k_{1}}z_{2}^{k_{2}}\dots z_{n}^{k_{n}}}

with a ≠ 0 {\displaystyle a\neq 0} is

N p ( x ) = log ⁡ | a | + k 1 x 1 + k 2 x 2 + ⋯ + k n x n . {\displaystyle N_{p}(x)=\log |a|+k_{1}x_{1}+k_{2}x_{2}+\cdots +k_{n}x_{n}.}

  • Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars. Vol. 35. Basel: Birkhäuser. ISBN 978-3-7643-8309-1. Zbl .
  • Viro, Oleg (2002), (PDF), Notices of the American Mathematical Society, 49 (8): 916–917.

Further reading

  • Theobald, Thorsten (2002). . Exp. Math. 11 (4): 513–526. doi:. Zbl .

External links