From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: <none>
Date: 19 Mar 1995 06:44:58 GMT

In article <1995Mar13.103745.2242@zh014.ubs.ubs.ch>,
 <Preda zhmhd. Mihailescu@zhflur.ubs.ch.ubs.com> wrote:
>I have this problem of concrete algebra:
>let p(x) and q(x) be polynomials with integer coefficients, irreducible over Z.
>let h(x) be the polynomials having all the possible products of roots of respectively one root of p and one of q. i.e., if p(x)=\prod_{i=1}^k (x-\alpha_i) and
>q(x)=\prod_{j=1}^l (x-\beta^j), then h(x)=\prod_{i=1,j=1}^{k,l} (x-\alpha_i * \beta_j).
>
>Now I want to express the polynomial h(x) in function of the coefficients of p and q.

Of course the coefficients of  h  are (up to sign) the elementary symmetric
functions of the products  \alpha_i*\beta_j. But rather than compute these
directly I would suggest computing the sums
	C_n = Sum_{i,j} (\alpha_i * \beta_j)^n.
Clearly  C_n = A_n*B_n  where each of  A_n  and  B_n  is a similar sum
computed from the  \alpha_i  and the  \beta_j. So the sequence of events 
would be: 
(1) given  p(x)  and  q(x), read off the symmetric functions  a_n  and  b_n
of their roots; 
(2) compute the terms A_n  and  B_n  using the Newton formulas:
	0= A_1 - a_1
	0= A_2 - a_1 A_1 + 2 a_2
	0= A_3 - a_1 A_2 + a_2 A_1 - 3 a_3
and so on. The  b_n  and  B_n  are similarly related.
(3) compute the terms  C_n = A_n * B_n
(4) use the Newton formulas again to compute the  c_n.

dave (one again championing the now-ignored Theory of Equations)

==============================================================================
Date: Tue, 5 Mar 1996 15:30:27 +0100
From: Daniel LAZARD <lazard@posso.ibp.fr>
To: rusin@math.niu.edu
Subject: product of roots

In polynomials/resultant.like, you give an answer for the problem of
finding the polynomial which has as roots all the products of one root
of a polynomial p by one root of a second polynomial q.

The simplest answer is 
        r(z) = resultant(p(x), x^degree(q)*q(z/x), x) More precisely,
if p and q are polynomials in x, r is obtained by following Maple
computation: 

resultant(p,normal(x^degree(q,x)*subs(x=z/x,q)),x)

Sincerely

                        Daniel Lazard

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