Southern California Math Mini-conference for Community College Students
INTRODUCTION TO
TROPICAL MATHEMATICS
Objectives

Origin of “Tropical” Mathematics

Applications

Definitions and Arithmetic operations

Monomials

The Freshman Dream

Possible research project
Origin of “Tropical” Mathematics
Origin of “Tropical” Mathematics
Brazilian dude Imre Simon
Applications of Tropical Mathematics
 Geometric Combinatorics
 Algebraic Geometry
 Mathematical Physics
 Number Theory
 Symplectic Geometry
 Computational Biology
Tropical Mathematics
Tropical mathematics is the study of the tropical
semiring:
(ℝ ⋃ {∞}, ⊕, ⊙)
Ring Definition
Ring: An algebraic structure consisting of:
A set
Together with two binary operations (usually
addition and multiplication)
With the following axioms:
Addition is commutative
Addition and multiplication are associative
Multiplication distributes over addition
There exists an additive identity
Each element in the set has an additive inverse
Semiring Definition
Semiring:
Similar to a ring, but without the requirement
that each element must have an additive
inverse.
Arithmetic operations
Tropical sum:
⊕ operator
x ⊕ y ≔ min{x, y}
1⊕2=1
5⊕8=5
(ℝ ⋃ {∞}, ⊕, ⊙)
// “tropical sum” operator
Arithmetic operations
Tropical product:
⊙ operator
x⊙y≔x+y
1⊙2=3
5 ⊙ 8 = 13
(ℝ ⋃ {∞}, ⊕, ⊙)
// “tropical product” operator
Arithmetic operations
Both addition and multiplication are commutative.
x⊕y=y⊕x
x⊙y=y⊙x
4⊕7=7⊕4
4=4
3⊙8=8⊙3
11 = 11
Arithmetic operations
Order of Operations is the same.
Distributive Law holds for both operations.
x ⊙ (y ⊕ z) = x ⊙ y ⊕ x ⊙ z
4 ⊙ (3 ⊕ 7) = 4 ⊙ 3 ⊕ 4 ⊙ 7
= 7 ⊕ 7
= 7
Arithmetic operations
Neutral Elements:
(ℝ ⋃ {∞}, ⊕, ⊙)
x+0=x
x·1=x
x⊕∞=x
4⊕∞=4
// ∞ is the additive identity.
x⊙0=x
4⊙0=4
// remember we’re adding here
Arithmetic operations
Additive Inverse:
The additive inverse, or opposite, of
a number a is the number that,
when added to a, yields zero.
Additive inverse of 7 is −7,
7 + (−7) = 0,
The additive inverse of a number is the number's
negative.
Arithmetic operations
Additive Inverse:
7 + -7 = 0
7 ⊕ -7 = -7
// min{7, -7}
Additive inverse does not exist for the tropical
semiring.
Arithmetic operations
Tropical difference:
11 ⊖ 3 = ?
This really means:
3 ⊕x =?
Change to: 11 ⊕ -3 = -3
(ℝ ⋃ {∞}, ⊕, ⊙)
// impossible.
// min{3, x}; no solution
// x is undefined
// illegal;
// no additive inverse
Tropical Semiring Definition
Tropical Semiring: (ℝ ⋃ {∞}, ⊕, ⊙) An algebraic structure
consisting of:
A set ℝ
Together with two binary operations ⊕, ⊙
With the following axioms:
Addition is commutative,
Addition and multiplication are associative,
Multiplication distributes over addition,
There exists an additive identity
No additive inverse
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Monomials
Let x₁, x₂, x₃, … , xn be variables that represent elements in the tropical
semiring (ℝ ⋃ {∞}, ⊕, ⊙).
A monomial is any product of these variables, where repetition is
allowed.
A monomial represents a function from ℝⁿ to ℝ.
x₂ ⊙ x₁ ⊙ x₃ ⊙ x₁ ⊙ x₄ ⊙ x₂ ⊙ x₃ ⊙ x₂ = x₁² x₂³ x₃² x₄
x₂ + x₁ + x₃ + x₁ + x₄ + x₂ + x₃ + x₂ = 2x₁ + 3x₂ + 2x₃ + x₄.
//= linear function
∷ Tropical monomials are linear functions with integer coefficients.
The Freshman Dream
Definition:
(x + y)³ = x³ + y³
//not true
(x + y)³ = x³ + x²y + xy² + y³
//true
But! Freshman Dream holds for Tropical Arithmetic
(x ⊕ y)³ = x³ ⊕ y³
//true
The Freshman Dream
Proof:
(x ⊕ y)³
= (x ⊕ y) ⊙ (x ⊕ y) ⊙ (x ⊕ y)
= (x ⊙ x ⊙ x) ⊕ (x ⊙ x ⊙ y) ⊕ (x ⊙ y ⊙ y) ⊕ (y ⊙ y ⊙ y)
= x³
⊕ x² y
⊕ xy²
⊕ y³
= x³ ⊕ y³
The Freshman Dream
Example:
(3 ⊕ 2)³ = 3³ ⊕ 2³
= (3 ⊕ 2) ⊙ (3 ⊕ 2) ⊙ (3 ⊕ 2)
= (3 ⊙ 3 ⊙ 3) ⊕ (3 ⊙ 3 ⊙ 2) ⊕ (3 ⊙ 2 ⊙ 2) ⊕ (2 ⊙ 2 ⊙ 2)
= 3³
⊕ 3² 2
⊕ 3 ⊙ 2²
⊕ 2³
= 3³
⊕ 2³
= 2³
=6
Research Problem
Research problem:
The tropical semiring generalizes to higher
dimensions: The set of convex polyhedra in ℝⁿ
can be made into a semiring by taking ⊙ as
“Minkowski sum” and ⊕ as “convex hull of the
union.” A natural subalgebra is the set of all
polyhedra that have a fixed recession cone C. If
n = 1 and C = ℝ≥0, this is the tropical semiring.
Develop linear algebra and algebraic geometry
over these semirings, and implement efficient
software for doing arithmetic with polyhedra
when n ≥ 2.