TITLE: Monic polynomial with integer coefficients with roots on unit circle, not roots of unity?
QUESTION [24 upvotes]: 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.

REPLY [4 votes]: This is an adelic comment based on algebraic dynamics. Each such polynomial $f(x)\in\mathbb{Z}[x]$ induces an algebraic dynamical system (i.e. an automorphism of a compact abelian group). If $f(x)$ has no roots that are roots of unity, then the system is known to have strictly positive entropy. The entropy is calculated by adding up the logs of the absolute values of the eigenvalues. If they are all on the unit circle then you'd get zero, which is wrong. So where is positive entropy coming from? It's from the $p$-adic eigenvalues of $f(x)$ (see Automorphisms of solenoids and p-adic entropy, Lind and Ward, Ergodic Th. & Dynam. Syst. 8 (1988), 411-419).
In other words, if $f(x)$ has all of its (complex) roots on the unit circle (and they are not roots of unity), then it must have $p$-adic roots having $p$-adic norm strictly greater than one. For the polynomial mentioned in the first sentence, there is a 5-adic root with norm > 1. In some sense, this is the correct (adelic) formulation of Kronecker's theorem mentioned above.