SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

Elementary symmetric polynomials. Viete’s Formulas.


Def. Symmetric polynomial. A function of two or more variables which remains unchanged under every interchange of two of the variables.


Example. xy + xz + yz



Def. Elementary symmetric polynomials. For the case of three variables x1, x2, x3, the elementary symmetric polynomials are

 

1)        σ1 = x1 + x2 + x3

             σ2 = x1 x2 + x1 x3 + x2 x3 

            σ3 = x1 x2 x3


which arise in the expansion

 

2)        (t - x1)(t - x2)(t - x3) = t3 - σ1t2 + σ2t - σ3



For the case of four variables x1, x2, x3, x4 the elementary symmetric polynomials are

 

3)        σ1 = x1 + x2 + x3 + x4

σ2 = x1 x2 + x1 x3 + x1 x4 + x2 x3 + x2 x4 + x3 x4

            σ3 = x1 x2 x3 +x1 x2 x4 + x1 x3 x4 + x2 x3 x4           (i.e. sum of all combinations of the variables x1, x2, ... , xn taken 3 at a time)

            σ4 = x1 x2 x3 x4


For the case of n variables x1, x2, ... , xn, the elementary symmetric polynomials are


4)        σ1 = x1 + x2 + ... + xn

            .......................

            σk = x1 x2 x3 ... + x1 x3 x4 ... + ...      (sum of all combinations of the variables x1, x2, ... , xn taken k at a time)

 

            .......................

            σn = x1 x2 x3 ... xn


which arise in the expansion

 

4)        (t - x1)(t - x2) ... (t - xn) = tn - σ1tn-1 + σ2tn-2 - σ2tn-3 + ... + (-1)nσn



Theorem. Any symmetric polynomial p(x1, x2, ... , xn) can be expressed as a polynomial in the elementary symmetric polynomials.


Examples.

 

1.         x2 + y2 = (x + y)2 - 2xy = σ12 - 2σ2             where σ1 = x + y and σ2 = xy

2.         x3 + y3 = (x + y)3 - 3xy(x + y) = σ112 - 3σ2) = σ13 - 3σ1σ2            where σ1 = x + y and σ2 = xy

 [Note. It can be shown through expansion that x3 + y3 = (x + y)3 - 3xy(x + y) ]

 


Def. The group of the polynomial. In the polynomial p(x1, x2, ... , xn), the set of all those permutations of the indices which leave the polynomial unchanged.




Viete’s Formulas. Let x1, x2, ... , xn be the n roots of the polynomial


            f(x) = xn + a1xn-1 + a2xn-2 + ... + an


where the coefficients a1, a2, ..., an are real or complex numbers. Then


            f(x) = (x - x1)(x - x2) ... (x - xn)


where x1, x2, ..., xn are real or complex numbers.


If we multiply out the expression (x - x1)(x - x2) ... (x - xn) we find


            -a1 = x1 + x2 + ... + xn

            a2 = x1 x2 + x1 x3 + x1 x4 + ...          (sum of all combinations of the variables x1, x2, ... , xn taken 2 at a time)

            -a3 = x1 x2 x3 + x1 x2 x4 + x1 x3 x4 ...         (sum of all combinations of the variables x1, x2, ... , xn taken 3 at a time)

            ............................................................

 

            -ak = x1 x2 x3... + x1 x3 x4 ... + x1 x3 x5 ...   (sum of all combinations of the variables x1, x2, ... , xn taken k at a time)

            ............................................................


 

            (-1)nan = x1 x2 x3 ... xn



Thus

 

a1 = -σ1

            a2 = σ2

              a3 = -σ3

            ..........

            an = (-1)nσn



Note. Note that the function

 

            f(x1, x2, ... , xn) = (x - x1)(x - x2) ... (x - xn)


is a symmetric function when regarded as a function of the roots x1, x2, ... , xn with x held constant.



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]