Re: [Axiom-math] Symmetric Functions

[Fabio S.]

> On 10/23/2014 01:45 PM, Fabio S. wrote:
> > Consider the following polynomial
> >
> > G := (y-(a*u+b*v))*(y-(a*v+b*u))
> >
> > It is symmetric both in (a,b) and (u,v). I would like to espress it as a
> > polynomial in Z[s,t,u,v,y]
> > where s=a+b and t=ab are the  symmetric elementary funcitions on a and b
> >
> > Is it possible in axiom?
> >
> > In other words, I am looking for a command which having G as input, returns
> >
> > y^2 - s*(u+v)*y + (s^2-2*t)u*v + t*(u^2+v^2)
>
> According to
> http://en.wikipedia.org/wiki/Symmetric_polynomial#Elementary_symmetric_polynomials
> the expression (u^2+v^2) doesn't look like an *elementary* symmetric
> polynomial in u and v.

Ralf, you are right of course. Maybe I didn't explain well what I looked 
for.
I want the result to be expressed in term of the elementary symmetric 
polynomials in {a,b}, nothing else: even if what we have is symmetric 
also in the {u,v} and hence it can be expressed also in the elementary 
symmetric polynomials in {u,v}, I am not interested in this last 
expression.
I am interested in trasforming an expression in
Z[a,b][u,v][y]
which is symmetric in a and b in an expression in
Z[s,t][u,v][y]

Anyway, your answer is great (and perfectly satisfactory): thanks a lot!

Fabio