Re: [Axiom-math] Symmetric Functions

[Raymond Rogers]

On 10/23/2014 10:08 AM, Ralf Hemmecke wrote:

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



(1) -> Z==>Integer; Q==>Fraction Z
                                                                   Type:
Void
(2) -> M==>DistributedMultivariatePolynomial([y], Q)
                                                                   Type:
Void
(3) -> F==>Fraction M
                                                                   Type:
Void
(4) -> P==>DistributedMultivariatePolynomial([u,v,a,b,s,t,p,q], F)
                                                                   Type:
Void
(5) -> g: P := (y-(a*u+b*v))*(y-(a*v+b*u))

         2           2        2                    2                       2
   (5)  u a b + u v a  + u v b  - y u a - y u b + v a b - y v a - y v b + y
Type:
DistributedMultivariatePolynomial([u,v,a,b,s,t,p,q],Fraction(DistributedMultivariatePolynomial([y],Fraction(Integer))))
(6) -> s1: P := a+b-s; s2: P := a*b-t; s3:P := u+v-p;s4:P:=u*v-q;

Type:
DistributedMultivariatePolynomial([u,v,a,b,s,t,p,q],Fraction(DistributedMultivariatePolynomial([y],Fraction(Integer))))
(7) -> G := groebner [g,s1,s2,s3,s4]

   (7)
                2                        2
   [u + v - p, v  - v p + q, a + b - s, b  - b s + t,
     2               2           2
    s q - y s p + t p  - 4t q + y ]
Type:
List(DistributedMultivariatePolynomial([u,v,a,b,s,t,p,q],Fraction(DistributedMultivariatePolynomial([y],Fraction(Integer)))))
(8) -> last(G)

         2               2           2
   (8)  s q - y s p + t p  - 4t q + y
Type:
DistributedMultivariatePolynomial([u,v,a,b,s,t,p,q],Fraction(DistributedMultivariatePolynomial([y],Fraction(Integer))))

Also check out:

(10) -> symFunc([a,b])$SymmetricFunctions(Polynomial(Integer))

   (10)  [1,b + a,a b]
                                            Type:
Vector(Polynomial(Integer))


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Good show!!!!!
The above summarizes and succinctly presents the "masters-thesis"
http://mattpap.github.io/masters-thesis/html/src/groebner.html
subsection:  Algebraic relations in invariant theory

It has a more itemized detailed approach but once the ideas are present the
above (Ralf) lays it out plainly; which, IMHO, the mattpap fails to do.

Ray