Date: Mon, 3 Apr 1995 18:35:40 -0400
To: rusin@math.niu.edu (Dave Rusin)
From: hshafey@julian.uwo.ca (Hany Shafey)
Subject: Re: Discriminant

>Well, what's to know? If P  is a polynomial with roots ri, the product
>S=Prod(ri-rj) is not well defined, since it depends on the order of the
>roots -- move them around and you may replace  S  by its negative
>(example - 2 roots => S = (r1-r2) = -(r2-r1). ) So we take  D = S^2;
>this is invariant under any permutation of the roots. In particular, it
>can be written as a polynomial in the coefficients of the polynomial  P.
>For P = ax^2+bx+c, D = b^2-4ac. It's not easy to compute the formula for
>the discriminant for a polynomial of large degree. One trick is to write
>the Vandermonde matrix, whose ij entry is ri^j. The determinant of this
>matrix is zero if any  ri = some other rj; comparing degrees convinces
>you the determinant is really just S. Then  D  is the determinant of 
>the square ofthis vandermonde thing; it's more instructive to look at
>V.transpose(V). Then you get sums of powers of the roots, which can
>be computed from the coefficients of  P.
>
>What else do you want to know?
>
>Thank you that is all!

==============================================================================

From: "Josep M. Lopez Besora" <jlopez@etse.urv.es>
Newsgroups: sci.math
Subject: Re: Discriminant
Date: 6 Apr 1995 15:55:31 GMT

hshafey@julian.uwo.ca (Hany Shafey) wrote:
>>Can anyone post some info on the topic of the Discriminant, I am 
having 
>trouble with it.
>
Well, I don't remember the exact formulation of the discriminant in this 
moment, but I remember that, up to some constant, the discriminant is 
the resultant of a polynomial and its derivative. That is, the 
discriminant is the result of computing a certain determinant made from 
the polynomial coefficients. In fact, I think you can construct a linear 
system this way:
take your n degree polynomial amd multiply it by an n-2 degree 
polynomial whose coefficients are the first n-1 unknowns, then take the 
n-1 degree derivative of the polynomial and multiply it by an n-1 degree 
of (again) unknown coefficients, add both results and ask this sum to be 
equal to the polynomial zero (that is, all coefficients equal to 0). You 
have a linear system with 2n-1 unknowns. The discriminant is (up to a 
constant) the determinant of this system.
Discriminant equal to zero is the necessary and sufficent condition for 
existence of multiple roots, as it is the existence of common roots for 
the polynomial and its derivative, which is equivalent to the 
possibility of expressing the zero polynomial as a combination of the 
polynomial and its derivative in the described way.