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The specific question is the following: I am given a set of $[L/2]$ numbers

$g(n) = \sqrt{ c(n)^2 + \alpha c(n) + \beta},$

where $c(n) = \cos(2\pi n/L)$ (so both $c(n)$ and $g(n)$ depend on $L$ too) and $n=0,1, \ldots, [L/2]-1$ with $L$ integer and $\alpha, \beta$ real. The question is: when are the $g(n)$ rationally independent? (with this I mean linearly independent over the rationals i.e. $\sum_n m(n) g(n) = 0$ with $m(n) \in \mathbb{Z}$ implies $m(n) =0$).

I guess the answer will be of the kind: for most values of $\alpha,\beta$ (i.e. except for a set of zero measure) and some $L$. The most important part is: what is the precise condition on $L$? Using Gauss result I can guess that the $c(n)$ are rationally independent for $L$ prime. Can this condition be relaxed given the functional form of $g(n)$?

Of course the question easily generalizes: given a set of numbers $c(n)$ and a function $f$ so that

$g(n) = f( c(n) ),$

what can I say about rational independence of $g(n)$ possibly respect to that of the $c(n)$? At least for some simple functions $f$.

Can you suggest me an approach to solve this problem?

Thanks to all

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    $I$ guess that the question can be a difficult one, so I am willing to accept as answer anything that can provide informations on how to solve or attack the problem.2011-04-09

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