Download Tropical Geometry Part 1 - Emory`s Math Department

AN EMORY MATH CIRCLE: INTRODUCTION TO TROPICAL GEOMETRY
JACKSON S. MORROW
1. W HAT IS T ROPICAL G EOMETRY ?
Geometry is a branch of mathematics concerned with questions of shape, size, relative
position of figures, and the properties of space. In your typical courses, you have studied
the geometry over the real numbers, which we denote by R, by considering graphs of
functions and curves in the Euclidean plane. For example, the function f ( x ) = x2 + 0.
Tropically, the graph of this function looks like
T
2
0
0
(´8, ´8)
T
F IGURE 1. The tropical picture of f ( x ) = x2 + 0
What happened to our function f ( x )? In the tropical world, the function became linear,
meaning that the graph of the function just looks like a bunch of line segments. During
this Math Circle, we will learn more about the tropical world and what tropical lines look
like. Roughly speaking, tropical geometry is the (piece-wise) linear version of geometry.
2. T ROPICAL A RITHMETIC
To begin our study of tropical geometry, we need to redefine how we conduct arithmetic. Up until this point, you have studied arithmetic in R. In this space, addition and
multiplication are what you know.
Brace yourself because this is all about to change when we take a trip down to the
tropics.
Date: February 14, 2016.
Many of the examples and problems for this Math Circle come from [MS15, Chapter 1.1].
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Definition 2.1 (Tropical Numbers). The tropical numbers are defined as so:
T = R Y t´8u .
This is just the real numbers R together with an extra point ´8. Visually, one should
think of T as:
t´8u
R
F IGURE 2. The tropical real line T
To understand why we define T as so, we need to discuss the arithmetic in the tropical
numbers.
Definition 2.2 (Tropical Arithmetic). In T, we define addition ‘ and multiplication d to
be:
x ‘ y := max( x, y)
x d y = x + y.
In words, the tropical sum of two numbers is their maximum, and the tropical product
of two numbers is their usual sum.
Example 2.2.1. Let’s do some basic examples:
5‘3 = 5
5d3 = 8
4‘9 = 9
4 d 9 = 13
3‘3 = 3
3 d 3 = 6.
Many of the familiar rules of arithmetic remain valid in the tropical world. For instance,
both addition and multiplication are commutative, so
x‘y = y‘x
x d y = y d x.
Also, the distributive law holds for tropical addition and multiplication
x d ( y ‘ z ) = ( x d y ) ‘ ( x d z ).
Example 2.2.2. Let’s do a basic example with the distributive laws:
3 d (7 ‘ 10) = 3 d 10 = 13
(3 d 7) ‘ (3 d 10) = 10 ‘ 13 = 13.
In usual arithmetic, we have the notion of an additive and multiplicative identity. In
particular, for any real number x, we know that 1 ¨ x = x and 0 + x = x. Ideally, we would
like to have an additive and multiplicative identity in the tropical world.
Definition 2.3. The tropical additive identity is the element t´8u of T since
´8 ‘ x = max(´8, x ) = x.
The tropical multiplicative identity is the element 0 of T since
0 d x = 0 + x = x.
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Question. Fill out the tropical addition and tropical multiplication tables. Do you see a
pattern?
‘
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
d
1
2
3
4
5
Question. Can you derive any different identities in tropical arithmetic? In particular,
compute
"
x‘0 =
Question. Prove the distributive law for tropical arithmetic. In particular, verify that
x + max(y, z) = max( x + y, x + z).
Hint: Do a few examples to convince yourself that this is true. Once you can see a pattern,
you can prove it. To prove this statement, you need to show that x + max(y, z) ě max( x +
y, x + z) and then that x + max(y, z) ď max( x + y, x + z).
Question. How would you define tropical division? Specifically, does there exists a number y such that x d y = 0.
Remark 2.4. One might think that tropical subtraction exists, however, this is not the case!
There is no real number x that we could call “4 minus 13” because the equation 13 ‘
x = 4 does not have a tropical solution, namely there is not a real number x such that
max(13, x ) = 4. This is a pretty weird thing, and it can/does cause problems later on.
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3. T ROPICAL L INES
To begin our discussion on tropical lines, we need to discuss tropical polynomials. Now
that we know about tropical arithmetic, our first example makes sense. Tropically, the
function f ( x ) looks like f trop ( x ) = x d x ‘ 0 = max(0, 2x ).
We want to make this process called tropicalization more systematic. Suppose that we
have a polynomial f ( x ) = an x n + an´1 x n´1 + ¨ ¨ ¨ + a1 x + a0 where n ą 0 and ai is a real
number for 0 ď i ď n. The tropicalization of f is defined by
f trop ( x ) = max( an + nx, an´1 + (n ´ 1) x, . . . , a1 + x, a0 ).
Example 3.0.1. Consider the polynomial g( x ) = 0 + x + (´1) x2 + (´3) x3 . The tropicalization of g( x ) is
gtrop ( x ) = max(0, x, ´1 + 2x, ´3 + 3x ).
How to graph tropical polynomials? Let’s consider the above polynomial g( x ) = 0 +
x + (´1) x2 + (´3) x3 . Above, we showed that gtrop ( x ) = max(0, x, ´1 + 2x, ´3 + 3x ). To
graph gtrop ( x ), we do the following:
(1) Draw the four lines: y = 0, y = x, y = ´1 + 2x, y = ´3 + 3x.
(2) The graph of gtrop ( x ) is given by the upper envelope of the lines.
‚ The upper envelope of the lines is defined as so: Imagine you have a infinitely
long string. If we pull the string down from above, then it will hit the lines
that we previously drew. The upper envelope is equal to the placement of the
string after we pull it down.
2
1
0
0
(´8, ´8)
1
2
F IGURE 3. The graph of gtrop ( x )
Question. Verify that Figure 1 is the graph of the function f trop ( x ) = max(0, 2x ).
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4. E XTRA Q UESTIONS
Question. Compute ( x ‘ y)2 ? Compute ( x ‘ y)3 ? Recall that the rows of Pascal’s triangle
are the coefficients appearing in the Binomial theorem. Using your above results, what
do you think the tropical version of Pascal’s triangle will look like? Remember that in
the tropical world, 0 is the multiplicative identity so the 1’s from Pascal’s triangle should
immediately be replaced by 0’s.
Question. The Freshman’s dream is the name given to the common mathematical error
that ( x + y)n = x n + yn . Using your above result show that in the tropical world, the
Freshmans’ dream comes true, namely show that
( x ‘ y)n = x n ‘ yn .
Question. If you have seen logarithms, can you come up with a formula for the tropicalization of a polynomial. In particular, if I give you the polynomial h( x ) = x3 + (´4) x2 +
(8) x + 3, how could you define htrop ( x ) using logarithms. Recall the rules of logarithms,
namely that
log(53 72 ) = log(53 ) + log(72 ) = 3 log(5) + 2 log(7).
R EFERENCES
[MS15] Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, vol. 161,
American Mathematical Soc., 2015.
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