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?



A transcendence question involving the exponential function
It is false. Equation $\sin x=1/x$ has infinitely many real solutions $(x\_n)$, take $z\_n=ix\_n$, we get $z\_n(e^{2z\_n}-1)=-2e^{z\_n}$.


