Let $P$ be a polynomial of fixed degree $d$ with integer coefficients of absolute values at most $n$. Assume that $P(\cos 2\pi/n)$ is no zero. Is there a lower bound for $|P(\cos 2\pi/n)|$ ? For instance, is this at least $n^{-C(d) }$ where $C(d)$ is some constant depending on $d$ ? 



Diophantine approximation
It seems one can get something by just looking at the Taylor expansion of $\cos$. Let me consider $\cos 1/n$ for simplicity. Plugging 


$$\cos 1/n = 1- (1/n)^2/2! + (1/n)^4/ 4! -.... $$


into $P= a\_d x^d + a\_{d-1} x^{d-1}+ ...+ a\_1 x + a\_0 $ one has 


$$(a\_d+ ...+ a\_0) - \frac{(1/n)^2}{2!} (da\_d +...+ a\_1) -....$$


If $a\_d+ ...+a\_0$ is not zero, it dominates everything else, as $|a\_i| \le n$. If it is zero, 
repeat the argument for $da\_d+ ...+a\_1$ and so on. By linear independency, this cannot go
for more than $d$ steps (I guess). So we should have a lower bound like $n^{-2d}$. 


