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.