Consider natural numbers as coefficients $a_i\in \mathbb{N}_0$.
Consider the polynomials $\sum_{i=0}^{m-1}a_iX^i + X^m$.
Note that $a_i\in \mathbb{N}_0$.
Question 1: What are classical results concerning these polynomials and their roots which distinguish them from all the other?
Consider the functions $f_{nm}(x) = nx\ \%\ m$ which sends $0 \leq x < m$ to the residue of $nx$ modulo $m$.
Consider the polynomials $\sum_{x=0}^{m-1}f_{nm}(x)X^x + X^m$.
Note that $f_{nm}(x)\in \mathbb{N}_0$.
Question 2: Is anything general known about (the roots of) these polynomials?
Consider the Fourier transform $\tilde{f}_{nm}(k)$ of $f_{nm}(x)$.
Consider the polynomials $\sum_{k=0}^{m-1}\tilde{f}_{nm}(k)X^k + X^m$.
Note that in general $\tilde{f}_{nm}(k)\in \mathbb{C}$.
Question 3: What can be said about these polynomials (if anything)?
Consider the $m$ roots $\zeta(k)$, $k = 0,\dots,m-1$, of any polynomial of degree $m$.
Consider the polynomials $\sum_{k=0}^{m-1}\zeta({\pi(k)})X^k + X^m$ for some permutation $\pi$ of the roots.
Note that in general $\zeta(k)\in \mathbb{C}$.
Question 4: What can be said about these polynomials (if anything)?
Consider the Fourier transform $\tilde{\zeta}(x)$ of the roots $\zeta(k)$ of any polynomial.
Consider the polynomials $\sum_{x=0}^{m-1}\tilde{\zeta}(\pi(x))X^x + X^m$ for some permutation $\pi$ of the FT.
Note that in general $\tilde{\zeta}(x)\in \mathbb{C}$.
Question 5: What can be said about these polynomials (if anything)?
For question 1: Since they are monic polynomials with coefficients in $\mathbb Z$, the roots are algebraic integers. Of course none of the roots are positive reals, nor do they have conjugates which are positive reals.