Re: Symmetric Functions
Raymond Rogers <[email protected]> Thu, 23 Oct 2014 15:03:01 -0400
| Newsgroups | gmane.comp.mathematics.axiom.general |
|---|---|
| Message-ID | <[email protected]> |
This is a multi-part message in MIME format. --===============9054822698290330351== Content-Type: multipart/alternative; boundary="------------020102040807040604070906" This is a multi-part message in MIME format. --------------020102040807040604070906 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 7bit 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)) > > > _______________________________________________ > Axiom-math mailing list > [email protected] > https://lists.nongnu.org/mailman/listinfo/axiom-math 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 --------------020102040807040604070906 Content-Type: text/html; charset=windows-1252 Content-Transfer-Encoding: 8bit <html> <head> <meta content="text/html; charset=windows-1252" http-equiv="Content-Type"> </head> <body bgcolor="#FFFFFF" text="#000000"> <br> <div class="moz-cite-prefix">On 10/23/2014 10:08 AM, Ralf Hemmecke wrote:<br> </div> <blockquote cite="mid:[email protected]" type="cite"> <pre wrap="">On 10/23/2014 01:45 PM, Fabio S. wrote: </pre> <blockquote type="cite"> <pre wrap=""> 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) </pre> </blockquote> <pre wrap=""> According to <a class="moz-txt-link-freetext" href="http://en.wikipedia.org/wiki/Symmetric_polynomial#Elementary_symmetric_polynomials">http://en.wikipedia.org/wiki/Symmetric_polynomial#Elementary_symmetric_polynomials</a> 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 <a class="moz-txt-link-abbreviated" href="mailto:[email protected]">[email protected]</a> <a class="moz-txt-link-freetext" href="https://lists.nongnu.org/mailman/listinfo/axiom-math">https://lists.nongnu.org/mailman/listinfo/axiom-math</a> </pre> </blockquote> Good show!!!!!<br> The above summarizes and succinctly presents the "masters-thesis"<br> <a class="moz-txt-link-freetext" href="http://mattpap.github.io/masters-thesis/html/src/groebner.html">http://mattpap.github.io/masters-thesis/html/src/groebner.html</a><br> subsection: <meta http-equiv="content-type" content="text/html; charset=windows-1252"> <meta http-equiv="content-type" content="text/html; charset=windows-1252"> Algebraic relations in invariant theory<br> <br> It has a more itemized detailed approach but once the ideas are present the<br> above (Ralf) lays it out plainly; which, IMHO, the mattpap fails to do.<br> <br> Ray<br> <br> <br> </body> </html> --------------020102040807040604070906-- --===============9054822698290330351== Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Disposition: inline _______________________________________________ Axiom-math mailing list [email protected] https://lists.nongnu.org/mailman/listinfo/axiom-math --===============9054822698290330351==--