Tropical Curves
Tropical Curves
Tropical mathematics has two operations, addition, , and multiplication, , which differ from traditional addition and subtraction. Tropical addition is defined as the minimum of the numbers while tropical multiplication is defined as the ordinary sum. For example, and . A power is repeated multiplication; for example, =x⊗x⊗x.
⊕
⊗
3⊕7=3
3⊗7=10
3
x
The graph (not shown here) of the tropical polynomial of one variable is simply the lower envelope of the lines , , , and .
p(x)=a⊗⊕b⊗⊕c⊗x⊕d
3
x
2
x
y=3x+a
y=2x+b
y=x+c
y=d
"A tropical polynomial function is given as the minimum of a finite set of linear functions. We define the hypersurface to be the set of all points at which this minimum is attained at least twice," in [1].
p:→R
n
ℋ(p)
x∈
n
For example, the surface is represented by the points where occurs in at least two of the terms.
(x⊗y)⊕⊗⊕⊗⊗6⊕⊗⊗2
7
y
2
x
2
y
3
x
-3
y
9
x
min(x+y,7y+2x,2y+3x+6,-3y+9x+2)
This Demonstration plots the surface of degree 6,
⊗⊗⊕⊗⊗⊕⊗⊗⊕⊗⊗⊕⊗⊗⊕⊗⊗
a
1
x
b
1
y
c
1
a
2
x
b
2
y
c
2
a
3
x
b
3
y
c
3
a
4
x
b
4
y
c
4
a
5
x
b
5
y
c
5
a
6
x
b
6
y
c
6
which is equivalent to plotting the points where at least two of the terms
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
a
4
b
4
c
4
a
5
b
5
c
5
a
6
b
6
c
6