As I am new to this forum, please correct me if this post is not appropriate. In that case I apologize.
Let $P(z)$ and $Q(z)$ be polynomials with coefficients in $\mathbb{C}$, furthermore let $Z(P)$ and $Z(Q)$ denote their zero sets. What can be said about $Z(P+Q)$?
Without imposing any further restrictions on $P$ and $Q$. I see that $Z(P) \cap Z(Q) \subset Z(P+Q)$. Or if we additionally assume that one of the polynomials dominates $P+Q$ in the sense that $|P(z)|\geq|P(z)+Q(z)|$ for all $z\in \mathbb{C}$, then clearly also $Z(P)\subset Z(P+Q)$ holds.
Without imposing to harsh restrictions (Very vague, I know) on the involved polynomials, what can be said?
Lastly, I really appreciate any help from you.