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 [email protected] https://lists.nongnu.org/mailman/listinfo/axiom-math
