Re: p-th power of quartic polynom of the n variables
- To: mathgroup at smc.vnet.net
- Subject: [mg67924] Re: p-th power of quartic polynom of the n variables
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Fri, 14 Jul 2006 02:11:24 -0400 (EDT)
- Organization: The University of Western Australia
- References: <e5os8c$hu5$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <e5os8c$hu5$1 at smc.vnet.net>, Michal Kvasnicka
<3.14159 at seznam.cz> wrote:
> I am looking for general formula of the polynom coeficients, which
> corresponds to the p-th (integer) power of the quartic polynom of n
> variables,
> Q(n,x) = a_ijkl*x_i*x_j*x_k*x_l + b_ijk*x_i*x_j*x_k + c_ij*x_i*x_j + d_i*x_i
> + e
> where Eisntein sumation convetion is apllied, i,j,k,l = 1, ..., n (n ...
> number of variables), a,b,c,d and e are real (or integer) coeficients
>
> I am looking for analytic expresions of the coeficients correponding to the
> expanded Q^p, where p is integer power.
Why? Do you think that such an expression will be useful?
It is not hard to compute such expression directly, e.g., with n=2, p=3,
With[{n = 2},
Sum[Subscript[a, i, j, k, l] *
Subscript[x, i] Subscript[x, j] Subscript[x, k] Subscript[x, l],
{i, n}, {j, n}, {k, n}, {l, n}] +
Sum[Subscript[b, i, j, k] *
Subscript[x, i] Subscript[x, j] Subscript[x, k],
{i, n}, {j, n}, {k, n}] +
Sum[Subscript[c, i, j] *
Subscript[x, i] Subscript[x, j], {i, n}, {j, n}] +
Sum[Subscript[d, i] Subscript[x, i], {i, n}] + e]
the coefficients you are after are
Collect[%^3, {Subscript[x, 1], Subscript[x, 2]}, Factor]
but I doubt that a general formula will be that useful in practical
computations.
Cheers,
Paul
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