There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example.


Now, for a monic polynomial of degree $n$, this is impossible (I think).


**So, my question is,** given a monic polynomial with integer coefficients of degree $n$,
what is the maximal number of roots that can lie on the unit circle, and not be roots of unity?


For example, $1 + 3 x + 3 x^2 + 3 x^3 + x^4$ has two roots on the unit circle, and two real roots.



Monic polynomial with integer coefficients with roots on unit circle, not roots of unity?
There exist irreducible monic polynomials such that all their roots apart from two lie on the unique circle (and are not roots of unity). Such polynomials can be chosen among Salem polynomials and they exit in arbitrary high degree. By definition a Salem polynomial $S(x)\in \mathbb Z[x]$ is a monic irreducible reciprocal polynomial with exactly two roots off the unit circle, both real and positive. Of course non of the roots of these polynomials are roots of unity, since these polynomials are irreducible.


See for example theorem 1.6 in the article


Automorphisms of even unimodular lattices and unramified Salem numbers of Gross and Mcmullen: 


<http://www.math.harvard.edu/~ctm/home/text/papers/unim/unim.pdf>


**Theorem.** For any odd integer $n\ge 3$ there exist infinitely many unramified Salem polynomials of degree $2n$.


