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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.

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    It seems unlikely that we could get a good characterization; for example, if $P=1$ and $Q=-1$ then $Z(P)=Z(Q)=\varnothing$ but $Z(P+Q)=\mathbb{C}$. Similarly if $Q=1-P$ then $Z(P)$ can be any finite subset of $\mathbb{C}$ but $Z(P+Q)=Z(1)=\varnothing$.2012-01-23
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    Thank you, that is a fair point. What about if assume that P,Q, P+Q are non-constant and their zero sets are nonempty. Is there anything interesting to be said?2012-01-23

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