Is there any sequence $ \{ Z\_{\nu} \}\_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z\_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in $Z\_{\nu}$ for all $\nu \in \mathbb{N}$ is identically zero?


Thank you!



A sequence that tell us if a holomorphic function of several variables is identically zero
Start from a countable dense subset $S$ of the unit ball, and take a sequence $Z\_\nu$ such that for all $s\in S$, one has $\nu Z\_\nu=s$ infinitely often. Then, any holomorphic function in $\mathbb{C}^n$ that vanishes along the sequence $Z\_\nu$ has in particular a subsequence of zeros accumulating to the origin that belong to the complex line generated by any $s\in S$. Therefore it vanishes identically on that complex line, by the principle of isolated zeros in one variable. Since the union of these lines is dense in $\mathbb{C}^n$, the function is identically zero by continuity. 


