interpolation properties of analytic paths Assume we are given $n$ points in $\mathbb{C}^k$ can we find an analytic path
$\phi:[0,1]\to \mathbb{C}^k$ passing through these $n$ points?

As Harald Hanche-Olsen pointed out, there is an interpolating polynomial of degree at most $n-1$ that does the job. Namely, let $p_1,\dots,p_n$ be the given points, pick any numbers $0\le t_1<t_2<\dots<t_n\le 1$, and define 
$$
\phi(t) = \sum_{j=1}^n p_j \frac{\prod_{l\ne j}(t-t_l)}{\prod_{l\ne j}(t_j-t_l)}
$$
This is a polynomial of degree at most $n-1$ such that $\phi(t_j)=p_j$ for all $j=1,\dots,n$.
