I know that the cyclotomic polynomial $\Phi_m = \prod (x-\zeta_m^c)$ for all $c$ coprime to $m$, with $\zeta_m$ a primitive $m^{th}$ root of unity is the unique minimal polynomial with integral coefficients for any $m^{th}$ root of unity. This often gives us negative integer coefficients, however, such as $\Phi_{15} = x^8-x^7+x^5-x^4+x^3-x+1$.
What is the minimal polynomial with nonnegative integral coefficients for any $m^{th}$ root of unity?
Or with coefficients in a given set such as $\{0,1\}$? It would have to be a multiple of $\Phi_m$ because $\mathbb{N} \subset \mathbb{Z}$, so would you approach by finding a polynomial to multiply with $\Phi_m$ to cancel all the negative terms?
I know there exist polynimials with nonnegative coeffecients that vanish for all roots $\zeta_m$ with $m>1$, for example $\sum\limits_{k=0}^{m-1}x^k$. This can be interpreted as the sum of all of the $m^{th}$ roots of unity, which is zero. For prime $m$ this is the minimal polynomial with nonnegative coefficients. For other $m$ it is not minimal, however. Another approach could be to factor these polynomials.