Non-trivial theorems on the roots of $P+Q$ for polynomials $P,Q$

Question

Let $P,Q$ be polynomials (with whatever coefficients you desire.) In general, the roots of $P+Q$ are not immediately calculable (except by direct computation) even if we have full information of $P$ and $Q$, including their roots.

My question: what are some non-trivial* theorems that give information on the roots of $P+Q$ given information on $P$ and $Q$, probably with certain restrictions? Ie. their location, sum, exact value(s)...anything!

Example: One theorem which fits the bill is Rouché's theorem, which can be applied to polynomials.

This question is, partly, a reference request.

*By non-trivial, I mean $P \neq-Q$, or things of that nature.

I am especially interested in the case that $P,Q$ both have roots strictly on the unit circle, and/or $P,Q$ restricted to $\mathbb{Z}[x]$.

Answer 1

This is just a comment that got too long:

One such result (though it may not be quite you want) is Hermite-Biehler theorem - $P,Q$ real coefficients have strictly interlacing zeroes iff $P+iQ$ has all zeroes in the (open) upper plane, or all zeroes in the (open) lower plane

(strictly interlacing means, $P,Q$ real polynomials with real zeroes which strictly alternate in order on the real line, one of $P$, then one of $Q$ etc, so in particular all zeroes are simple - by Rolle, $P,P'$ are like that for any $P$ with simple real roots)

Then there are inversion results also - eg $P$ degree $n$ and $P^*$ its inversion (reverse and conjugate coefficients - $P(z)=\sum_{0 \le k \le n} a_kz^k, a_n \ne 0, P^*(z)=\sum_{0 \le k \le n} \bar a_kz^{n-k}$, then $P$ has all roots in the closed unit disc or all roots outside the closed unit disc iff $P+\alpha P^*$ has roots on the unit circle only for all $|\alpha|=1$