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