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

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 $$\mathbb{Z}[x]$$.

• Well, the sum of roots can found by Vieta, but that's pretty trivial too. – Ivan Neretin Jul 16 '20 at 6:58

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 $$|\alpha|=1$$