As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

## Sunday, July 8, 2007

### Polynomials

From the Fundamental Theorem of Algebra we know that an n-th degree univariate polynomial has n complex roots.

An example for the case n=3, let x^3+b*x^2+c*x+d=0 have roots r1, r2 and r3. Then

x^3+b*x^2+c*x+d=(x-r1)*(x - r2)*(x - r3),

expanding the right side of this equation gives

x^3+b*x^2+c*x+d=x^3-(r1+r2+r3)*x^2+(r1*r2+r1*r3+r2*r3)*x-r1*r2*r3.

This shows that the coefficients of an n-th degree univariate polynomial are multivariate polynomials in its roots.

Because changing the order of the roots does not change the coefficients the polynomials

-(r1+r2+r3),

(r1*r2+r1*r3+r2*r3) and

-r1*r2*r3,

are symmetric, and are called s(1), s(2) and s(3), or the elementary symmetric functions for n=3.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)