Set associated with a complex-valued polynomial
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.}
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 .
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