This question roughly translates to seeing if the intersection of two limits of sets, is the same as the limit of the intersections. This is of course false in general, but what conditions needs to be fulfilled for this to be true? This is the setting I am interested in:
Suppose we have two sequences of polynomials, $P_n(x,y)$ and $Q_n(x,y)$ in two complex variables.
Suppose that for each fixed $y^\star$, set of zeros of $P_n(x,y^\star)$ converge to a set $C(y^\star)$, in the following sense: Each point $x \in C(y^\star)$ is a limit of a subsequence of zeros of $P_n(x,y^\star)$ as $n \rightarrow \infty$.
Similarly, for each fixed $x^\star$, set of zeros of $Q_n(x^\star,y)$ converge to $C(x^\star)$. Thus, for each fixed $(x^\star,y^\star)$, we have two limit sets $C(x^\star)$ and $C(y^\star).$
Now, let $C$ be the set of points $(x,y) \in \mathbb{C}^2$ such that $x \in C(y)$ and $y \in C(x).$
Under what conditions do the zeros of the system $P_n(x,y)=Q_n(x,y)=0$ converge (in the above sense) to $C$?
