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[57]:= 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[58]= {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