Let $(z_n)$ be a sequence of complex numbers satisfying $|z_n|\to +\infty$ and such that $\{e^{z_n}\mid n \in \mathbb{N}\}$ is infinite.

Is it always true that $\{(z_n,e^{z_n})\mid n \in\mathbb{N}\}$ is Zariski-dense in $\mathbb{C}^2$? In other words, if $p(x,y) \in \mathbb{C}[x,y]$ is a polynomial such that $p(z_n,e^{z_n})=0$ for every $n \in \mathbb{N}$, is it always the case that $p = 0$?

This is clearly true if $z_n$ are taken to be real numbers by an "order of growth" argument. But in general I'm not even completely sure if this should be true. Any ideas?

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