Is there a polynomial $f \in \Bbb C[x_1,...,x_n]$ such that $f(v_i)$ is nonzero for all $i$ and $f(0)=0$?

Question

Suppose $v_1,\dots,v_k$ are finitely many nonzero vectors in $\Bbb C^n$. Is there a polynomial $f \in \Bbb C[x_1,...,x_n]$ such that $f(v_i)$ is nonzero for all $i$ and $f(0)=0$?

In the real case, the polynomial $f=x_1^2+\cdots+x_n^2$ satisfies the condition, but the complex case does not seem trivial.

Any hints?

Answer 1

For each $v_l$, choose a nonzero coordinate $v_{l,i}$. Then define $f(x)=\prod_l{(v_{l,i}-x_i)}-\prod_l{v_{l,i}}$.