On the Quadratic Equation

Bernd SturmfelsUC Berkeley & MPI Leipzig

An Invitation to Non-Linear Algebra

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Back in Ninth GradeA quadratic equation has the form

ax2 + bx + c = 0.

The letter x is the unknown.

The three quantities a, b, c are parameters. In applications, theyare measurements from an experiment. They change many times.

How do we solve this equation?

The teacher presents a general formula.The students memorize that formula.

Why do we solve this equation?No clue.

The curriculum requires it.

Math class is totally boring....2 / 21

The Formula

A general quadratic equation ax2 + bx + c = 0 has two solutions:

x =−b ±

√b2 − 4ac

2a.

The discriminant is the expression

D = b2 − 4ac

There is a case distinction concerning the nature of the solution:

D > 0 oder D = 0 oder D < 0.

There are almost always two complex solutions.

The number of real solutions is

Two or one or zero.

3 / 21

Completing the SquareDerivation: The following equations are all equivalent:

ax2 + bx + c = 0

ax2 + bx = −c

ax2 + bx + b2

4a = b2

4a − c

a (x + b2a)2 = b2−4ac

4a

(x + b2a)2 = b2−4ac

4a2

x + b2a = ±

√b2−4ac2a

x = − b2a + ±

√b2−4ac2a

What to do with polynomials in x of higher degree?Numerical solutions, symbolic representations of the roots, ....

What to do with polynomials in several unknowns?Grobner bases, resultants, discriminants, draw pictures, ....

4 / 21

QuotesMathematics is the language in which God has written theuniverse. Galileo Galilei

Beauty is the first test: there is no permanent place in the worldfor ugly mathematics. Godfrey Harold Hardy

Mathematicians are like Frenchmen: whatever you say to themthey translate into their own language and forthwith it issomething entirely different. Johann Wolfgang von Goethe

5 / 21

ElegantMathematics, rightly viewed, possesses not only truth, but supremebeauty. Bertrand Russell

x2 + y 2 + z2 − 2xyz − 1 = 0

Oliver Labs: surfex.AlgebraicSurface.net Nina’s Fashion Show

[Jiawang Nie, Kristian Ranestad, B. St: The algebraic degree of semidefinite programming, Math Progr. (2010)]

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Useful

04/06/15 06:21New Mathematics Could Neutralize Pathogens That Resist Antibiotics - Scientific American

Page 1 of 4http://www.scientificamerican.com/article/new-mathematics-could-neutralize-pathogens-that-resist-antibiotics/

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A new algorithm that deciphers how bacteria genesgain resistance to antibiotics could help improvedrug cycling regimes.

Credit: NIAID / Flickr

Health » News

New Mathematics Could NeutralizePathogens That Resist AntibioticsA “time machine” algorithm, backed by experimental data, reveals how to cycle drugs to

reverse resistance

By Sarah Lewin | May 26, 2015

Bacteria that make us sick are bad enough,but many of them also continually evolve inways that help them develop resistance tocommon antibiotic drugs, making ourmedications less effective or even moot.Doctors try to reduce the evolution bycycling through various drugs over time,hoping that as resistance develops to one,the increased use of a new drug or thewidespread reuse of an old drug will catchsome of the bugs off guard.

The plans for cycling drugs are not thatscientific, however, and don’t always workefficiently, allowing bacteria to continue todevelop resistance. Now a new algorithmthat deciphers how bacteria genes createresistance in the first place could greatlyimprove such a plan. The “time machine”software, developed by biologists andmathematicians, could help reverse resistant mutations and render the bacteriavulnerable to drugs again.

Miriam Barlow, a biologist from the University of California, Merced, first hit on theidea while trying to predict how antibiotic resistance would evolve several years ago.But she lacked the experimental data or the mathematics to quantify it. “We werepushing evolution forward, trying to predict how antibiotic resistance would evolve,and we saw a lot of trade-offs,” Barlow says. Introducing an antibiotic might lead tobacteria developing resistance but it might also lead to them losing resistance to some

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Tomorrow's Medicine

04/06/15 06:09New Mathematics Could Neutralize Pathogens That Resist Antibiotics - Scientific American

Page 1 of 4http://www.scientificamerican.com/article/new-mathematics-could-neutralize-pathogens-that-resist-antibiotics/

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Sign In | Register 0

Search ScientificAmerican.com

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Subscribe to All Access »

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Give a Gift »

View the Latest Issue »

Subscribe News & Features Topics Blogs Videos & Podcasts Education Citizen Science SA Magazine SA Mind Books SA en español

A new algorithm that deciphers how bacteria genesgain resistance to antibiotics could help improvedrug cycling regimes.

Credit: NIAID / Flickr

Health » News

New Mathematics Could NeutralizePathogens That Resist AntibioticsA “time machine” algorithm, backed by experimental data, reveals how to cycle drugs to

reverse resistance

By Sarah Lewin | May 26, 2015

Bacteria that make us sick are bad enough,but many of them also continually evolve inways that help them develop resistance tocommon antibiotic drugs, making ourmedications less effective or even moot.Doctors try to reduce the evolution bycycling through various drugs over time,hoping that as resistance develops to one,the increased use of a new drug or thewidespread reuse of an old drug will catchsome of the bugs off guard.

The plans for cycling drugs are not thatscientific, however, and don’t always workefficiently, allowing bacteria to continue todevelop resistance. Now a new algorithmthat deciphers how bacteria genes createresistance in the first place could greatlyimprove such a plan. The “time machine”software, developed by biologists andmathematicians, could help reverse resistant mutations and render the bacteriavulnerable to drugs again.

Miriam Barlow, a biologist from the University of California, Merced, first hit on theidea while trying to predict how antibiotic resistance would evolve several years ago.But she lacked the experimental data or the mathematics to quantify it. “We werepushing evolution forward, trying to predict how antibiotic resistance would evolve,and we saw a lot of trade-offs,” Barlow says. Introducing an antibiotic might lead tobacteria developing resistance but it might also lead to them losing resistance to some

1 :: Email :: Print

Follow Us:

More from Scientific American

ADVERTISEMENT

More on this Topic

Tomorrow's Medicine

7 / 21

Surprising Addition Multiplication3⊕ 8 = 3 3� 8 = 118⊕ 3 = 3 8� 3 = 115⊕ 0 = 0 5� 0 = 5

5⊕∞ = 5 5�∞ =∞Rows in Pascal’s Triangle are the coefficients of (x ⊕ y)�n

00 0

0 0 00 0 0 0

0 0 0 0 0

(0�x ⊕ 0� y)�(0�x ⊕ 0�y)� · · · � (0�x ⊕ 0�y)e

d+yf+x

c+2ya+2x b+x+y

8 / 21

Surprising Addition Multiplication3⊕ 8 = 3 3� 8 = 118⊕ 3 = 3 8� 3 = 115⊕ 0 = 0 5� 0 = 5

5⊕∞ = 5 5�∞ =∞Rows in Pascal’s Triangle are the coefficients of (x ⊕ y)�n

00 0

0 0 00 0 0 0

0 0 0 0 0

(0�x ⊕ 0� y)�(0�x ⊕ 0�y)� · · · � (0�x ⊕ 0�y)e

d+yf+x

c+2ya+2x b+x+y

9 / 21

VarietiesThe set of solutions to a system of polynomial equationsin n variables is called a variety in Rn.

Example: Quadratic curves in the plane (n = 2):

a · x2 + b · xy + c · y2 + d · x + e · y + f = 0.

Two quadratic equations in x und y ...10/06/16 08:11220px-Is-circle-ellipse-s.svg.png 220×160 pixels

Page 1 of 1https://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Is-circle-ellipse-s.svg/220px-Is-circle-ellipse-s.svg.png

... have almost always four complex solutions. [Bezout 1764]

The discriminant is a huge polynomial in the coefficients.It specifies the case distinction: 0,1,2,3 or 4 real solutions.

10 / 21

My Favorite VarietyThe set of solutions to the equation

x4 + y4 + z4 − x2y2 − x2z2 − y2z2 − x2 − y2 − z2 + 1 = 0

is a surface of degree four in R3 that has 16 singular points:

[Ernst Eduard Kummer, 1810-1893]

Kummer surfaces have applications in cryptography.11 / 21

StatisticsWatching too much soccer on TV leads to hair loss?In a study, 296 people were asked about their hair length and howmany hours per week they watch soccer on TV. The data:

U =

full hair medium little hair

≤ 2 hours 51 45 332–6 hours 28 30 29≥ 6 hours 15 27 38

Is there a correlation between watching soccer and hair loss?

Extra info: Our study involved 126 men and 170 women:

U =

3 9 154 12 207 21 35

+

48 36 1824 18 98 6 3

We cannot reject the null hypothesis:

Hair length is conditionally independent of soccer on TV given gender.

12 / 21

StatisticsWatching too much soccer on TV leads to hair loss?In a study, 296 people were asked about their hair length and howmany hours per week they watch soccer on TV. The data:

U =

full hair medium little hair

≤ 2 hours 51 45 332–6 hours 28 30 29≥ 6 hours 15 27 38

Is there a correlation between watching soccer and hair loss?

Extra info: Our study involved 126 men and 170 women:

U =

3 9 154 12 207 21 35

+

48 36 1824 18 98 6 3

We cannot reject the null hypothesis:

Hair length is conditionally independent of soccer on TV given gender.13 / 21

Algebraic StatisticsPhilosophy: Statistical models are algebraic varieties.

Conditional independence of two ternary random variables:

This is the cubic hypersurface in R9 defined by

det

p11 p12 p13p21 p22 p23p31 p32 p33

= 0.

Given any data matrix (uij), one seeks to maximize thelikelihood function pu1111 pu1212 · · · p

u3333 over all points in this model.

This leads to a system of polynomial equations. It has almostalways 10 complex solutions. The discriminant is a polynomialin the data u11, u12, . . . , u33. It specifies the case distinction.

[J. Hauenstein, J. Rodriguez, B. St: Maximum likelihood for matrices with rank constraints, J. Alg. Stat (2014)][J. Rodriguez, X. Tang: Data discriminants of likelihood equations, ISSAC 2015]

14 / 21

More SoccerBy flattening, we can turn a soccer ball into a square:

[K. Kubjas, P. Parrilo, B. St.: How to flatten a soccer ball, 2017]15 / 21

Algebraic Formulation

Given: Three polynomials f , g , h in three variables x , y , z .

Example: f = xy + xz + yz , g = xyz , h = 1− x2 − y2 − z2

Desired: The image of “soccerball”

B ={

(x , y , z) ∈ R3 : h(x , y , z) ≥ 0}

under the flattening map

φ : R3 → R2, (x , y , z) 7→(f (x , y , z), g(x , y , z)

).

Question: For which parameters (a, b) ∈ R2 do the equations

f (x , y , z)− a = g(x , y , z)− b = h(x , y , z) = 0

have a solution (x , y , z) in B ? Discriminant? Case Distinction?

16 / 21

Picture 1f = xy + xz + yz , g = xyz , h = 1− x2 − y2 − z2

-0.5 0.5 1.0

-0.2

-0.1

0.1

0.2

17 / 21

Picture 2f = xy + xz + yz , g = xyz , h = 1− x2 − y2 − z2

-0.5 0.0 0.5 1.0

-0.2

-0.1

0.0

0.1

0.2

18 / 21

The 3-Ellipse

What was that picture on the title page?

This variety is an algebraic curve of degree 8.If we vary the radius then we obtain a surface of degree 8.

Discriminant? Case Distinction?19 / 21

Conclusion

Very important lessons can be learned from the formula

x =−b ±

√b2 − 4ac

2a.

I Many phenomena in the sciences and engineering can bedescribed by a mathematical model like ax2 + bx + c = 0.

I Realistic models are non-linear and algebraic.

I Models depend on parameters, like a, b, and c .

I The parameters can be measured, but data are noisy.

I Solving a non-linear model requires algebraic geometry.

I The solution is an algebraic function. It is piecewise smooth.

I Number of real (positive) solutions depends on data a, b, c .

I There is a case distinction, characterized by the discriminant.

20 / 21

Journal

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