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From: | Raymond Rogers |
Subject: | Re: [Axiom-math] Symmetric Functions |
Date: | Thu, 23 Oct 2014 15:03:01 -0400 |
User-agent: | Mozilla/5.0 (X11; Linux x86_64; rv:31.0) Gecko/20100101 Thunderbird/31.2.0 |
On 10/23/2014 10:08 AM, Ralf Hemmecke
wrote:
Good show!!!!!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)) _______________________________________________ Axiom-math mailing list address@hidden https://lists.nongnu.org/mailman/listinfo/axiom-math 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 |
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