       Re: Help generalizing Liouville's Polynomial Identity

• To: mathgroup at smc.vnet.net
• Subject: [mg103648] Re: [mg103596] Help generalizing Liouville's Polynomial Identity
• From: "Kurt TeKolste" <tekolste at fastmail.us>
• Date: Thu, 1 Oct 2009 06:39:24 -0400 (EDT)
• References: <200909291140.HAA25752@smc.vnet.net>

```It looks as if you will need to go quite a ways to the find the needed
number of variables:

In:= nextP[n_] :=
Total[Flatten[
Table[(x[i] + signj x[j] + signk x[k] + signl x[l])^4, {i, 1,
n - 3}, {j, i + 1, n - 2}, {k, j + 1, n - 1}, {l, j + 1,
n}, {signj, {-1, 1}}, {signk, {-1, 1}}, {signl, {-1, 1}}]]]
IrreduciblePolynomialQ[nextP[#]] & /@ Range[1, 20]

Out= {False, False, False, True, True, True, True, True, True, \
True, True, True, True, True, True, True, True, True, True, True}

ekt

On Tue, 29 Sep 2009 07:40 -0400, "TPiezas" <tpiezas at gmail.com> wrote:
> Hello all,
>
> "Liouville's polynomial identity" is given by
> http://mathworld.wolfram.com/LiouvillePolynomialIdentity.html and can
> be concisely encoded as,
>
> 6(x1^2 + x2^2 + x3^2 + x4^2)^2 = Sum(x_i +/- x_j)^4
>
> To determine the number of terms of the summation, since we are to
> choose 2 objects from 4, then this is Binomial[4,2] = 6. But as there
> are 2 sign changes, then total is 2 x 6 = 12 terms, given explicitly
> in the link above.  Going higher, and choosing 3 objects out of n
> variables, I found that,
>
> 60(x1^2 + x2^2 + ... + x7^2)^2 = Sum(x_i +/- x_j +/- x_k)^4
>
> is true. The RHS contains 4 x 35 = 140 terms. (If the x_i are non-
> zero, one cannot express the square of the sum of less than 7 squares
> in a similar manner.)
>
> Question: If we take the next step,
>
> a(x1^2 + x2^2 + ... + x_n^2)^2 = Sum(x_i +/- x_j +/- x_k +/- x_m)^4
>
> for some positive integer "a", then what is the least n, if we are to
> choose 4 objects at a time out of the x_n?
>
> I believe the RHS is a simple matter for Mathematica to calculate, and
> one can incrementally test n = 5,6,7,...etc until a neat identity is
> found.
>
> - Titus
>
>
>
Regards,
Kurt Tekolste

```

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