Let $n>0$ be an even integer divisible by $4$


Let $R(t)=r\_0+r\_1t+ \cdots + r\_{n-1}t^{n-1}$


be a polynomial with nonzero integer coefficients in $\{-1,1\}$
such that


$R(\omega)$ is a nonzero integer for all complex $\omega$ $\notin$ $\{-1,1\}$
such that
$$
\omega^n=1
$$


Can we deduce that all these integers $R(\omega)$ have the same sign ???



sign of values of an integer polynomial on roots of unity
$(1+x+x^2+x^3+x^4+x^5+ \ldots + x^{n-1})-2x^{n/2}$. 


EDIT: It is customary after someone answers your question not to change the question. As it happens, the slightest possible change to my answer *still* provides an answer to the latest version of your question (at the time of this edit). The little icon produced for my username (produced completely randomly as it happens) reflects my opinion of the practice of changing the question after it has been answered.


