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
(
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.}
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.}
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})}
complex numbers with values in the Euclidean space
R
n
,
{\displaystyle \mathbb {R} ^{n},}
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}
amoeba
A
p
{\displaystyle {\mathcal {A}}_{p}}
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 .[1]
Let
V
⊂
(
C
∗
)
n
{\displaystyle V\subset (\mathbb {C} ^{*})^{n}}
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}}
a
j
∈
C
{\displaystyle a_{j}\in \mathbb {C} }
z
j
=
z
1
j
1
⋯
z
n
j
n
{\displaystyle z^{j}=z_{1}^{j_{1}}\cdots z_{n}^{j_{n}}}
z
=
(
z
1
,
…
,
z
n
)
{\displaystyle z=(z_{1},\dots ,z_{n})}
j
=
(
j
1
,
…
,
j
n
)
{\displaystyle j=(j_{1},\dots ,j_{n})}
Δ
f
{\displaystyle \Delta _{f}}
Newton polyhedron of
f
{\displaystyle f}
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}}
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
∖
A
p
{\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}}
#
(
Δ
f
∩
Z
n
)
{\displaystyle \#(\Delta _{f}\cap \mathbb {Z} ^{n})}
Δ
f
{\displaystyle \Delta _{f}}
[2]
There is an injection from the set of connected components of complement
R
n
∖
A
p
{\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}}
Δ
f
∩
Z
n
{\displaystyle \Delta _{f}\cap \mathbb {Z} ^{n}}
Δ
f
{\displaystyle \Delta _{f}}
R
n
∖
A
p
{\displaystyle \mathbb {R} ^{n}\setminus {\mathcal {A}}_{p}}
Δ
f
{\displaystyle \Delta _{f}}
[2]
If
V
⊂
(
C
∗
)
2
{\displaystyle V\subset (\mathbb {C} ^{*})^{2}}
A
p
(
V
)
{\displaystyle {\mathcal {A}}_{p}(V)}
π
2
Area
(
Δ
f
)
{\displaystyle \pi ^{2}{\text{Area}}(\Delta _{f})}
[2]
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}
x
=
(
x
1
,
x
2
,
…
,
x
n
)
.
{\displaystyle x=(x_{1},x_{2},\dots ,x_{n}).}
N
p
{\displaystyle N_{p}}
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)}
[3]
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}
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 et al (2007) p. 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 . 
![{\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},}](http://wikimedia.org/api/rest_v1/media/math/render/svg/907b8ad5fef02069b4d36cc3379b402080d5143b)
