Re: Symmetric polynomials
- To: mathgroup at smc.vnet.net
- Subject: [mg68995] Re: [mg68940] Symmetric polynomials
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Sat, 26 Aug 2006 02:04:35 -0400 (EDT)
- References: <200608250934.FAA09161@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
shubi at nusun.jinr.ru wrote: > Dear All, > > Is the possibility in "Mathematica" express symmetric functions, for > example: > > P=y1^2 y2 y3 + y1 y2^2 y3 + y1 y2 y3^2 + y1^2 y2 y4 + > y1 y2^2 y4 + y1^2 y3 y4 + y2^2 y3 y4 + y1 y3^2 y4 + > y2 y3^2 y4 + y1 y2 y4^2 + y1 y3 y4^2 + y2 y3 y4^2; > > by the standard symmetric polynomials: > S1=y1+y2+y3+y4; > S2=y1^2+y2^2+y3^2+y4^2; > S3=y1^3+y2^3+y3^3+y4^3; > . . . > > Best regards > Nodar Shubitidze > Joint Institute for Nuclear Research > Dubna, Moscow region, Russia Yes, but you will need quartics as well. A reasonable way to do this is via PolynomialReduce. You will define polynomial relations for your "s" variables rather than setting them explicitly to polynomials. poly = y1^2*y2*y3 + y1*y2^2*y3 + y1*y2*y3^2 + y1^2*y2*y4 + y1*y2^2*y4 + y1^2*y3*y4 + y2^2*y3*y4 + y1*y3^2*y4 + y2*y3^2*y4 + y1*y2*y4^2 + y1*y3*y4^2 + y2*y3*y4^2; s[j_] := Apply[Plus,Variables[poly]^j] gb = GroebnerBasis[{s1-s[1], s2-s[2], s3-s[3], s4-s[4]}, Variables[poly]]; In[35]:= InputForm[Last[PolynomialReduce[poly, gb, Variables[poly]]]] Out[35]//InputForm= (s1^2*s2 - s2^2 - 2*s1*s3 + 2*s4)/2 Daniel Lichtblau Wolfram Research
- References:
- Symmetric polynomials
- From: shubi@nusun.jinr.ru
- Symmetric polynomials