From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: why can't you factorize this polynomial?
Date: 26 Oct 1995 20:43:55 GMT
In article ,
Gregor Stoelting wrote:
>E(n,k):= n^2 + k^2 +1
>
>P(n,k):=
>
> l j-1 l
>---- ---- ----
>/ sigma_j(n) || E(n-i, k) || E(n-i, k-1)
>\ || ||
>---- ---- ----
>j= 0 i=0 i=j
>
>where sigma_j are nonzero constants according to k.
>Is it possible to proof, that this polymomial
>doen't have nontrivial factors in Q[k]?
Well, it has the factor E(n-j,k-1). Perhaps you meant for the last
product to be taken over i=j,...,l-1 ?
Even then it seems you must know something about your sigma_j.
What if all sigma_j(n)=0 except for one of them? Then P is clearly
a product. A less trivial example occurs with l=1, sigma_0=1,
sigma_1=-2/3. Then P(-1,k)=E(1,k-1)+(-2/3)E(1,k)= (1/3)(k-1)(k-5).
dave