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Added "Tropical Polynomials" handout

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Mark 2025-01-22 12:31:47 -08:00
parent 7597343fe3
commit 3b41ea714a
Signed by: Mark
GPG Key ID: C6D63995FE72FD80
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280
src/Advanced/Tropical Polynomials/handout.typ Executable file
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/// If false, hide instructor info.
///
/// Compile with the following command to hide solutions:
/// `typst compile main.typ --input show_solutions=false`
///
/// Solutions are shown by default. This behavior
/// is less surprising than hiding content by default.
#let show_solutions = {
if "show_solutions" in sys.inputs {
// Show solutions unless they're explicitly disabled
not (
sys.inputs.show_solutions == "false" or sys.inputs.show_solutions == "no"
)
} else {
// Show solutions by default
true
}
}
// Colors
#let ored = rgb("D62121")
#let ogrape = rgb("9C36B5")
#let ocyan = rgb("2288BF")
#let oteal = rgb("12B886")
#let ogreen = rgb("37B26D")
#let oblue = rgb("1C7ED6")
//
// MARK: header
//
#let make_title(
group,
quarter,
title,
subtitle,
) = {
align(
center,
block(
width: 60%,
height: auto,
breakable: false,
align(
center,
stack(
spacing: 7pt,
(
text(size: 10pt, group) + h(1fr) + text(size: 10pt, quarter)
),
line(length: 100%, stroke: 0.2mm),
(
text(size: 20pt, title) + linebreak() + text(size: 10pt, subtitle)
),
line(length: 100%, stroke: 0.2mm),
),
),
),
)
}
#let warn = {
set text(ored)
align(
center,
block(
width: 60%,
height: auto,
breakable: false,
fill: rgb(255, 255, 255),
stroke: ored + 2pt,
inset: 3mm,
(
align(center, text(weight: "bold", size: 12pt, [Instructor's Handout]))
+ parbreak()
+ align(
left,
text(
size: 10pt,
[This handout contains solutions and notes.]
+ linebreak()
+ [Recompile without solutions before distributing.],
),
)
),
),
)
}
#let preparedby(name) = (
text(
size: 10pt,
[Prepared by ]
+ name
+ [ on ]
+ datetime
.today()
.display("[month repr:long] [day padding:none], [year]"),
)
)
//
// MARK: Solutions
//
#let solution(content) = {
if show_solutions {
align(
center,
stack(
block(
width: 100%,
breakable: false,
fill: ored,
stroke: ored + 2pt,
inset: 1.5mm,
(
align(left, text(fill: white, weight: "bold", [Solution:]))
),
),
block(
width: 100%,
height: auto,
breakable: false,
fill: ored.lighten(80%).desaturate(10%),
stroke: ored + 2pt,
inset: 3mm,
align(left, content),
),
),
)
}
}
#let notsolution(content) = {
if not show_solutions { content }
}
//
// MARK: Sections
//
#let generic(t) = block(
above: 8mm,
below: 2mm,
text(weight: "bold", t),
)
#let _generic_base(kind, ..args) = {
counter("obj").step()
if args.pos().len() == 0 {
generic([
#kind
#context counter("obj").display():
])
} else {
generic(
[
#kind
#context counter("obj").display():
]
+ " "
+ args.pos().at(0),
)
}
}
#let problem(..args) = _generic_base("Problem", ..args)
#let definition(..args) = _generic_base("Definition", ..args)
#let theorem(..args) = _generic_base("Theorem", ..args)
//
// MARK: Misc
//
#let hint(content) = {
text(fill: rgb(100, 100, 100), style: "oblique", "Hint: ")
text(fill: rgb(100, 100, 100), content)
}
#let note(content) = {
text(fill: rgb(100, 100, 100), content)
}
#let examplesolution(content) = {
let c = oblue
align(
center,
stack(
block(
width: 100%,
breakable: false,
fill: c,
stroke: c + 2pt,
inset: 1.5mm,
(
align(left, text(fill: white, weight: "bold", [Example solution:]))
),
),
block(
width: 100%,
height: auto,
breakable: false,
fill: c.lighten(80%).desaturate(10%),
stroke: c + 2pt,
inset: 3mm,
align(left, content),
),
),
)
}
//
// MARK: wrapper
//
#let handout(
doc,
group: none,
quarter: none,
title: none,
by: none,
subtitle: none,
) = {
set par(leading: 0.55em, first-line-indent: 0mm, justify: true)
set text(font: "New Computer Modern")
set par(spacing: 0.5em)
show list: set block(spacing: 0.5em, below: 1em)
set heading(numbering: (..nums) => nums.pos().at(0))
set page(
margin: 20mm,
width: 8.5in,
height: 11in,
footer: align(
center,
context counter(page).display(),
),
footer-descent: 5mm,
)
set list(
tight: false,
indent: 5mm,
spacing: 3mm,
)
show heading.where(level: 1): it => {
set align(center)
set text(weight: "bold")
block[
Section #counter(heading).display(): #text(it.body)
]
}
make_title(
group,
quarter,
title,
{
if by == none { none } else { [#preparedby(by)\ ] }
if subtitle == none { none } else { subtitle }
},
)
if show_solutions {
warn
}
doc
}

70
src/Advanced/Tropical Polynomials/macros.typ Normal file
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#import "./handout.typ": *
#import "@preview/cetz:0.3.1"
// Shorthand, we'll be using these a lot.
#let tp = sym.plus.circle
#let tm = sym.times.circle
#let graphgrid(inner_content) = {
align(
center,
box(
inset: 3mm,
cetz.canvas({
import cetz.draw: *
let x = 5.25
grid(
(0, 0), (x, x), step: 0.75,
stroke: luma(100) + 0.3mm
)
if (inner_content != none) {
inner_content
}
mark((0, x + 0.5), (0, x + 1), symbol: ">", fill: black, scale: 1)
mark((x + 0.5, 0), (x + 1, 0), symbol: ">", fill: black, scale: 1)
line(
(0, x + 0.25),
(0, 0),
(x + 0.25, 0),
stroke: 0.75mm + black,
)
}),
),
)
}
/// Adds extra padding to an equation.
/// Used as follows:
///
/// #eqmbox($
/// f(x) = -2(x #tp 2)(x #tp 8)
/// $)
///
/// Note that there are newlines between the $ and content,
/// this gives us display math (which is what we want when using this macro)
#let eqnbox(eqn) = {
align(
center,
box(
inset: 3mm,
eqn,
),
)
}
#let dotline(a, b) = {
cetz.draw.line(
a,
b,
stroke: (
dash: "dashed",
thickness: 0.5mm,
paint: ored,
),
)
}

22
src/Advanced/Tropical Polynomials/main.typ Normal file
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#import "./handout.typ": *
#show: doc => handout(
doc,
group: "Advanced 2",
quarter: link(
"https://betalupi.com/handouts",
"betalupi.com/handouts",
),
title: [Tropical Polynomials],
by: "Mark",
subtitle: "Based on a handout by Bryant Mathews",
)
#include "parts/00 arithmetic.typ"
#pagebreak()
#include "parts/01 polynomials.typ"
#pagebreak()
#include "parts/02 cubic.typ"

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src/Advanced/Tropical Polynomials/meta.toml Normal file
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[metadata]
title = "Tropical Polynomials"
[publish]
handout = true
solutions = true

293
src/Advanced/Tropical Polynomials/parts/00 arithmetic.typ Normal file
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#import "../handout.typ": *
#import "../macros.typ": *
= Tropical Arithmetic
#definition()
The _tropical sum_ of two numbers is their minimum:
$
x #tp y = min(x, y)
$
#definition()
The _tropical product_ of two numbers is their sum:
$
x #tm y = x + y
$
#problem()
- Is tropical addition commutative? \
#note([i.e, does $x #tp y = y #tp x$?])
- Is tropical addition associative? \
#note([i.e, does $(x #tp y) #tp z = x #tp (y #tp z)$?])
- Is there a tropical additive identity? \
#note([i.e, is there an $i$ so that $x #tp i = x$ for all real $x$?])
#solution([
- Is tropical addition commutative?\
Yes, $min(min(x,y),z) = min(x,y,z) = min(x,min(y,z))$
- Is tropical addition associative? \
Yes, $min(x,y) = min(y,x)$
- Is there a tropical additive identity? \
No. There is no $n$ where $x <= n$ for all real $x$
])
#v(1fr)
#problem()
Let's expand $#sym.RR$ to include a tropical additive identity.
- What would be an appropriate name for this new number?
- Give a reasonable definition for...
- the tropical sum of this number and a real number $x$
- the tropical sum of this number and itself
- the tropical product of this number and a real number $x$
- the tropical product of this number and itself
#solution([
#sym.infinity makes sense, with
$#sym.infinity #tp x = x$; #h(1em)
$#sym.infinity #tp #sym.infinity = #sym.infinity$; #h(1em)
$#sym.infinity #tm x = #sym.infinity$; #h(1em) and
$#sym.infinity #tm #sym.infinity = #sym.infinity$
])
#v(1fr)
#pagebreak() // MARK: page
#problem()
Do tropical additive inverses exist? \
#note([
Is there an inverse $y$ for every $x$ so that $x #tp y = #sym.infinity$?
])
#solution([
No. Unless $x = #sym.infinity$, there is no x where $min(x, y) = #sym.infinity$
])
#v(1fr)
#problem()
Is tropical multiplication associative? \
#note([Does $(x #tm y) #tm z = x #tm (y #tm z)$ for all $x,y,z$?])
#solution([Yes, since (normal) addition is associative])
#v(1fr)
#problem()
Is tropical multiplication commutative? \
#note([Does $x #tm y = y #tm x$ for all $x, y$?])
#solution([Yes, since (normal) addition is commutative])
#v(1fr)
#problem()
Is there a tropical multiplicative identity? \
#note([Is there an $i$ so that $x #tm i = x$ for all $x$?])
#solution([Yes, it is 0.])
#v(1fr)
#problem()
Do tropical multiplicative inverses always exist? \
#note([
For every $x != #sym.infinity$, does there exist an inverse $y$ so that $x #tm y = i$, \
where $i$ is the additive identity?
])
#solution([Yes, it is $-x$. For $x != 0$, $x #tm (-x) = 0$])
#v(1fr)
#pagebreak() // MARK: page
#problem()
Is tropical multiplication distributive over addition? \
#note([Does $x #tm (y #tp z) = x #tm y #tp x #tm z$?])
#solution([Yes, $x + min(y,z) = min(x+y, x+z)$])
#v(1fr)
#problem()
Fill the following tropical addition and multiplication tables
#let col = 10mm
#notsolution(
table(
columns: (1fr, 1fr),
align: center,
stroke: none,
table(
columns: (col, col, col, col, col, col),
align: center,
table.header(
[$#tp$],
[$1$],
[$2$],
[$3$],
[$4$],
[$#sym.infinity$],
),
box(inset: 3pt, $1$), [], [], [], [], [],
box(inset: 3pt, $2$), [], [], [], [], [],
box(inset: 3pt, $3$), [], [], [], [], [],
box(inset: 3pt, $4$), [], [], [], [], [],
box(inset: 3pt, $#sym.infinity$), [], [], [], [], [],
),
table(
columns: (col, col, col, col, col, col),
align: center,
table.header(
[$#tm$],
[$0$],
[$1$],
[$2$],
[$3$],
[$4$],
),
box(inset: 3pt, $0$), [], [], [], [], [],
box(inset: 3pt, $1$), [], [], [], [], [],
box(inset: 3pt, $2$), [], [], [], [], [],
box(inset: 3pt, $3$), [], [], [], [], [],
box(inset: 3pt, $4$), [], [], [], [], [],
),
),
)
#solution(
table(
columns: (1fr, 1fr),
align: center,
stroke: none,
table(
columns: (col, col, col, col, col, col),
align: center,
table.header(
[$#tp$],
[$1$],
[$2$],
[$3$],
[$4$],
[$#sym.infinity$],
),
box(inset: 3pt, $1$),
box(inset: 3pt, $1$),
box(inset: 3pt, $1$),
box(inset: 3pt, $1$),
box(inset: 3pt, $1$),
box(inset: 3pt, $1$),
box(inset: 3pt, $2$),
box(inset: 3pt, $1$),
box(inset: 3pt, $2$),
box(inset: 3pt, $2$),
box(inset: 3pt, $2$),
box(inset: 3pt, $2$),
box(inset: 3pt, $3$),
box(inset: 3pt, $1$),
box(inset: 3pt, $2$),
box(inset: 3pt, $3$),
box(inset: 3pt, $3$),
box(inset: 3pt, $3$),
box(inset: 3pt, $4$),
box(inset: 3pt, $1$),
box(inset: 3pt, $2$),
box(inset: 3pt, $3$),
box(inset: 3pt, $4$),
box(inset: 3pt, $4$),
box(inset: 3pt, $#sym.infinity$),
box(inset: 3pt, $1$),
box(inset: 3pt, $2$),
box(inset: 3pt, $3$),
box(inset: 3pt, $4$),
box(inset: 3pt, $#sym.infinity$),
),
table(
columns: (col, col, col, col, col, col),
align: center,
table.header(
[$#tm$],
[$0$],
[$1$],
[$2$],
[$3$],
[$4$],
),
box(inset: 3pt, $0$),
box(inset: 3pt, $0$),
box(inset: 3pt, $1$),
box(inset: 3pt, $2$),
box(inset: 3pt, $3$),
box(inset: 3pt, $4$),
box(inset: 3pt, $1$),
box(inset: 3pt, $1$),
box(inset: 3pt, $2$),
box(inset: 3pt, $3$),
box(inset: 3pt, $4$),
box(inset: 3pt, $5$),
box(inset: 3pt, $2$),
box(inset: 3pt, $2$),
box(inset: 3pt, $3$),
box(inset: 3pt, $4$),
box(inset: 3pt, $5$),
box(inset: 3pt, $6$),
box(inset: 3pt, $3$),
box(inset: 3pt, $3$),
box(inset: 3pt, $4$),
box(inset: 3pt, $5$),
box(inset: 3pt, $6$),
box(inset: 3pt, $7$),
box(inset: 3pt, $4$),
box(inset: 3pt, $4$),
box(inset: 3pt, $5$),
box(inset: 3pt, $6$),
box(inset: 3pt, $7$),
box(inset: 3pt, $8$),
),
),
)
#v(2mm)
#problem()
Expand and simplify $f(x) = (x #tp 2)(x #tp 3)$, then evaluate $f(1)$ and $f(4)$ \
Adjacent parenthesis imply tropical multiplication
#solution([
$
(x #tp 2)(x #tp 3)
&= x^2 #tp 2x #tp 3x #tp (2 #tm 3) \
&= x^2 #tp (2 #tp 3)x #tp (2 #tm 3) \
&= x^2 #tp 2x #tp 5
$
Also, $f(1) = 2$ and $f(4) = 5$.
])
#v(1fr)

387
src/Advanced/Tropical Polynomials/parts/01 polynomials.typ Normal file
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#import "../handout.typ": *
#import "../macros.typ": *
#import "@preview/cetz:0.3.1"
= Tropical Polynomials
#definition()
A _polynomial_ is an expression formed by adding and multiplying numbers and a variable $x$. \
Every polynomial can be written as
#align(
center,
box(
inset: 3mm,
$
c_0 + c_1 x + c_2 x^2 + ... + c_n x^n
$,
),
)
for some nonnegative integer $n$ and coefficients $c_0, c_1, ..., c_n$. \
The _degree_ of a polynomial is the largest $n$ for which $c_n$ is nonzero.
#theorem()
The _fundamental theorem of algebra_ implies that any non-constant polynomial with real coefficients
can be written as a product of polynomials of degree 1 or 2 with real coefficients.
#v(2mm)
For example, the polynomial $-160 - 64x - 2x^2 + 17x^3 + 8x^4 + x^5$ \
can be written as $(x^2 + 2x+5)(x-2)(x+4)(x+4)$
#v(2mm)
A similar theorem exists for polynomials with complex coefficients. \
These coefficients may be found using the roots of this polynomial. \
As it turns out, there are formulas that determine the roots of quadratic, cubic, and quartic #note([(degree 2, 3, and 4)]) polynomials. There are no formulas for the roots of polynomials with larger degrees---in this case, we usually rely on approximate roots found by computers.
#v(2mm)
In this section, we will analyze tropical polynomials:
- Is there a fundamental theorem of tropical algebra?
- Is there a tropical quadratic formula? How about a cubic formula?
- Is it difficult to find the roots of tropical polynomials with large degrees?
#definition()
A _tropical_ polynomial is a polynomial that uses tropical addition and multiplication. \
In other words, it is an expression of the form
#align(
center,
box(
inset: 3mm,
$
c_0 #tp (c_1 #tm x) #tp (c_2 #tm x^2) #tp ... #tp (c_n #tm x^n)
$,
),
)
where all exponents represent repeated tropical multiplication.
#pagebreak() // MARK: page
#problem()
Draw a graph of the tropical polynomial $f(x) = x^2 #tp 1x #tp 4$. \
#hint([$1x$ is not equal to $x$.])
#notsolution(graphgrid(none))
#solution([
$f(x) = min(2x , 1+x, 4)$, which looks like:
#graphgrid({
import cetz.draw: *
let step = 0.75
dotline((0, 0), (4 * step, 8 * step))
dotline((0, 1 * step), (7 * step, 8 * step))
dotline((0, 4 * step), (8 * step, 4 * step))
line(
(0, 0),
(1 * step, 2 * step),
(3 * step, 4 * step),
(7.5 * step, 4 * step),
stroke: 1mm + oblue,
)
})
])
#problem()
Now, factor $f(x) = x^2 #tp 1x #tp 4$ into two polynomials with degree 1. \
In other words, find $r$ and $s$ so that
#align(
center,
box(
inset: 3mm,
$
x^2 #tp 1x #tp 4 = (x #tp r)(x #tp s)
$,
),
)
we will call $r$ and $s$ the _roots_ of $f$.
#solution([
Because $(x #tp r)(x #tp s) = x^2 #tp (r #tp s)x #tp s r$, we must have $r #tp s = 1$ and $r #tm s = 4$. \
In standard notation, we need $min(r, s) = 1$ and $r + s = 4$, so we take $r = 1$ and $s = 3$:
#v(2mm)
$
f(x) = x^2 #tp 1x #tp 4 = (x #tp 1)(x #tp 3)
$
])
#v(1fr)
#problem()
How can we use the graph to determine these roots?
#solution([The roots are the corners of the graph.])
#v(0.5fr)
#pagebreak() // MARK: page
#problem()
Graph $f(x) = -2x^2 #tp x #tp 8$. \
#hint([Use half scale. 1 box = 2 units.])
#notsolution(graphgrid(none))
#solution([
#graphgrid({
import cetz.draw: *
let step = 0.75
dotline((0, 0), (8 * step, 8 * step))
dotline((0.5 * step, 0), (4 * step, 8 * step))
dotline((0, 4 * step), (8 * step, 4 * step))
line(
(0.5 * step, 0),
(1 * step, 1 * step),
(4 * step, 4 * step),
(7.5 * step, 4 * step),
stroke: 1mm + oblue,
)
})
])
#problem()
Find a factorization of $f$ in the form $a(x #tp r)(x#tp s)$.
#solution([
We (tropically) factor out $-2$ to get
#eqnbox($
f(x) = -2(x^2 #tp 2x #tp 10)
$)
by the same process as the previous problem, we get
#eqnbox($
f(x) = -2(x #tp 2)(x #tp 8)
$)
])
#v(1fr)
#problem()
Can you see the roots $r$ and $s$ in the graph? \
How are the roots related to the coefficients of $f$? \
#hint([look at consecutive coefficients: $0 - (-2) = 2$])
#solution([
The roots are the differences between consecutive coefficients of $f$:
- $0-(-2) = 2$
- $8 - 0 = 8$
])
#v(0.5fr)
#problem()
Find a tropical polynomial that has roots $4$ and $5$ \
and always produces $7$ for sufficiently large inputs.
#solution([
We are looking for $f(x) = a x^2 #tp b x #tp c$. \
Since $f(#sym.infinity) = 7$, we know that $c = 7$. \
Using the pattern from the previous problem, we'll subtract $5$ from $c$ to get $b = 2$, \
and $4$ from $b$ to get $a = -2$.
And so, $f(x) = -2x^2 #tp 2x #tp 7$
#v(2mm)
Subtracting roots in the opposite order does not work.
])
#v(1fr)
#pagebreak() // MARK: page
#problem()
Graph $f(x) = 1x^2 #tp 3x #tp 5$.
#notsolution(graphgrid(none))
#solution([
The graphs of all three terms intersect at the same point:
#graphgrid({
import cetz.draw: *
let step = 0.75
dotline((0, 1 * step), (3.5 * step, 8 * step))
dotline((0, 5 * step), (8 * step, 5 * step))
dotline((0, 3 * step), (5 * step, 8 * step))
line(
(0, 1 * step),
(2 * step, 5 * step),
(7.5 * step, 5 * step),
stroke: 1mm + oblue,
)
})
])
#problem()
Find a factorization of $f$ in the form $a(x #tp r)(x#tp s)$.
#solution(
eqnbox($
f(x) = 1x^2 #tp 3 x #tp 5 = 1(x #tp 2)^2
$),
)
#v(1fr)
#problem()
How is this graph different from the previous two? \
How is this polynomial's factorization different from the previous two? \
How are the roots of $f$ related to its coefficients?
#solution([
The factorization contains the same term twice. \
Also note that the differences between consecutive coefficients of $f$ are both two.
])
#v(0.5fr)
#pagebreak() // MARK: page
#problem()
Graph $f(x) = 2x^2 #tp 4x #tp 4$.
#notsolution(graphgrid(none))
#solution(
graphgrid({
import cetz.draw: *
let step = 0.75
dotline((0, 2 * step), (3 * step, 8 * step))
dotline((0, 4 * step), (5 * step, 8 * step))
dotline((0, 4 * step), (8 * step, 4 * step))
line(
(0, 2 * step),
(1 * step, 4 * step),
(7.5 * step, 4 * step),
stroke: 1mm + oblue,
)
}),
)
#problem()
Find a factorization of $f$ in the form $a(x #tp r)(x#tp s)$, or show that one does not exist.
#solution([
We can factor out a 2 to get $f(x) = 2(x^2 #tp 2x #tp 2)$,
but $x^2 #tp 2x #tp 2$ does not factor. \
There are no $a$ and $b$ with minimum 2 and sum 2.
])
#v(1fr)
#problem()
Find a polynomial that has the same graph as $f$, but can be factored.
#solution([
$
2x^2 #tp 3x #tp 4 = 2(x #tp 1)^2
$
])
#v(1fr)
#pagebreak() // MARK: page
#theorem()
The _fundamental thorem of tropical algebra_ states that for every tropical polynomial $f$, \
there exists a _unique_ tropical polynomial $accent(f, macron)$ with the same graph that can be factored \
into linear factors.
#v(2mm)
Whenever we say "the roots of $f$", we really mean "the roots of $accent(f, macron)$." \
$f$ and $accent(f, macron)$ might be the same polynomial.
#problem()
If $f(x) = a x^2 #tp b x #tp c$, then $accent(f, macron)(x) = a x^2 #tp B x #tp c$ for some $B$. \
Find a formula for $B$ in terms of $a$, $b$, and $c$. \
#hint([there are two cases to consider.])
#solution([
If we want to factor $a(x^2 #tp (b-a)x #tp (c-a))$, we need to find $r$ and $s$ so that
- $min(r,s) = b-a$, and
- $r + s = c - a$
#v(2mm)
This is possible if and only if $2(b-a) <= c-a$, \
or equivalently if $b <= (a+c) #sym.div 2$
#v(8mm)
*Case 1:* If $b <= (a + c #sym.div) 2$, then $accent(f, macron) = f$ and $b = B$.
#v(2mm)
*Case 2:* If $b > (a + c #sym.div) 2$, then
$
accent(f, macron)(x)
&= a x^2 #tp ((a+c)/2)x #tp c \
&= a(x #tp (c-a)/2)^2
$
has the same graph as $f$, and thus $B = (a+c) #sym.div 2$
#v(8mm)
We can combine these results as follows:
$
B = min(b, (a+c)/2)
$
])
#v(1fr)
#problem()
Find a tropical quadratic formula in terms of $a$, $b$, and $c$ \
for the roots $x$ of a tropical polynomial $f(x) = a x^2 #tp b x #tp c$. \
#hint([
again, there are two cases. \
Remember that "roots of $f$" means "roots of $accent(f, macron)$".
])
#solution([
*Case 1:* If $b <= (a+c) #sym.div 2$, then $accent(f, macron) = f$ has roots $b-a$ and $c-b$, so
$
accent(f, macron)(x) = a(x #tp (b-a))(x #tp (c-b))
$
#v(8mm)
*Case 2:* If $b > (a+c) #sym.div 2$, then $accent(f, macron)$ has root $(c-a) #sym.div$ with multiplicity 2, so
$
accent(f, macron)(x) = a(x #tp (c-a)/2)^2
$
#v(8mm)
It is interesting to note that the condition $2b < a+ c$ for there to be two distinct roots becomes $b^2 > a c$ in tropical notation. This is reminiscent of the discriminant condition for standard polynomials!
])
#v(1fr)

212
src/Advanced/Tropical Polynomials/parts/02 cubic.typ Normal file
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@ -0,0 +1,212 @@
#import "../handout.typ": *
#import "../macros.typ": *
#import "@preview/cetz:0.3.1"
= Tropical Cubic Polynomials
#problem()
Consider the polynomial $f(x) = x^3 #tp x^2 #tp 3x #tp 6$. \
- sketch a graph of this polynomial
- use this graph to find the roots of $f$
- write (and expand) a product of linear factors with the same graph as $f$.
#notsolution(graphgrid(none))
#solution([
- Roots are 1, 2, and 3.
- $accent(f, macron)(x) = x^3 #tp 1x^2 #tp 3x #tp 6 = (x #tp 1)(x #tp 2)(x #tp 3)$
#graphgrid({
import cetz.draw: *
let step = 0.75
dotline((0, 0), (2.66 * step, 8 * step))
dotline((0, 1 * step), (3.5 * step, 8 * step))
dotline((0, 3 * step), (5 * step, 8 * step))
dotline((0, 6 * step), (8 * step, 6 * step))
line(
(0, 0),
(1 * step, 3 * step),
(2 * step, 5 * step),
(3 * step, 6 * step),
(7.5 * step, 6 * step),
stroke: 1mm + oblue,
)
})
])
#v(1fr)
#pagebreak() // MARK: page
#problem()
Consider the polynomial $f(x) = x^3 #tp x^2 #tp 6x #tp 6$. \
- sketch a graph of this polynomial
- use this graph to find the roots of $f$
- write (and expand) a product of linear factors with the same graph as $f$.
#notsolution(graphgrid(none))
#solution([
- Roots are 1, 2.5, and 2.5.
- $accent(f, macron)(x) = x^3 #tp 1x^2 #tp 3.5x #tp 6 = (x #tp 1)(x #tp 2.5)^2$
#graphgrid({
import cetz.draw: *
let step = 0.75
dotline((0, 0), (2.66 * step, 8 * step))
dotline((0, 1 * step), (3.5 * step, 8 * step))
dotline((0, 6 * step), (2 * step, 8 * step))
dotline((0, 6 * step), (8 * step, 6 * step))
line(
(0, 0),
(1 * step, 3 * step),
(2.5 * step, 6 * step),
(7.5 * step, 6 * step),
stroke: 1mm + oblue,
)
})
])
#v(1fr)
#problem()
Consider the polynomial $f(x) = x^3 #tp 6x^2 #tp 6x #tp 6$. \
- sketch a graph of this polynomial
- use this graph to find the roots of $f$
- write (and expand) a product of linear factors with the same graph as $f$.
#notsolution(graphgrid(none))
#solution([
- Roots are 2, 2, and 2.
- $accent(f, macron)(x) = x^3 #tp 2x^2 #tp 4x #tp 6 = (x #tp 2)^3$
#graphgrid({
import cetz.draw: *
let step = 0.75
dotline((0, 0), (2.66 * step, 8 * step))
dotline((0, 6 * step), (1 * step, 8 * step))
dotline((0, 6 * step), (2 * step, 8 * step))
dotline((0, 6 * step), (8 * step, 6 * step))
line(
(0, 0),
(2 * step, 6 * step),
(7.5 * step, 6 * step),
stroke: 1mm + oblue,
)
})
])
#v(1fr)
#pagebreak() // MARK: page
#problem()
If $f(x) = a x^3 #tp b x^2 #tp c x #tp d$, then $accent(f, macron)(x) = a x^3 #tp B x^2 #tp C x #tp d$ for some $B$ and $C$. \
Using the last three problems, find formulas for $B$ and $C$ in terms of $a$, $b$, $c$, and $d$.
#solution([
$
B = min(b, (a+c)/2, (2a+d)/2)
$
$
C = min(c, (b+d)/2, (a+2d)/2)
$
])
#v(1fr)
#pagebreak() // MARK: page
#problem()
What are the roots of the following polynomial?
#align(
center,
box(
inset: 3mm,
$
3 x^6 #tp 4 x^5 #tp 2 x^4 #tp x^3 #tp x^2 #tp 4 x #tp 5
$,
),
)
#solution([
We have
$
accent(f, macron)(x) = 3x^6 #tp 2x^5 #tp 1x^4 #tp x^3 #tp 1x^2 #tp 3x #tp 5
$
which has roots $-1$, $-1$, $-1$, $1$, $2$, $2$
])
#v(1fr)
#pagebreak() // MARK: page
#problem()
If
$
f(x) = c_0 #tp c_1 x #tp c_2 x^2 #tp ... #tp c_n x^n
$
then
$
accent(f, macron)(x) = c_0 #tp C_1 x #tp C_2 x^2 #tp ... #tp C_(n-1) x^(n-1) #tp c_n x^n
$
#v(2mm)
Find a formula for each $C_i$ in terms of $c_0, c_1, ..., c_n$.
#solution([
$
A_j
&= min_(l<=j<k)( (a_l - a_k) / (k-l) (k-j) + a_k ) \
&= min_(l<=j<k)( a_l (k-j) / (k-l) + a_k (j-l) / (k-l) )
$
#v(2mm)
which is a weighted average of some $a_l$ and $a_k$, with $l<=j<k$
])
#v(1fr)
#problem()
With the same setup as the previous problem, \
find formulas for the roots $r_1, r_2, ..., r_n$.
#solution([
The roots are the differences between consecutive coefficients of $accent(f, macron)$:
$
r_i = A_i - A_(i-1)
$
where we set $A_n = a_n$ and $A_0 = a_0$.
])
#v(1fr)
#problem()
Can you find a geometric interpretation of these formulas \
in terms of the points $(-i, c_i)$ for $0 <= i <= n$?
#solution([
The inequality #note([(for $l <= k < k$)])
$
A_j <= (a_l - a_k) / (k-l) (k-j)+a_k
$
states that the point $(-j,A_j)$ must lie on or below the line segment between the points $(-k, a_k)$ and $(-l, a_l)$.
This makes it easy to find the $A_j$ using a graph of the points $(-i, a_i)$ for $0 <= i <=n$.
])
#v(0.5fr)