Let D be the unit disk, and $d\ge 2$. $F=(f_1,...f_d)$ is a holomorphic map over the closure of $D^d$ to $C^d$. What information can we get about the image of $\{0\}$x $D^{d-1}$ under $F$?
Hope some characterization about it. Is it locally contained in a zero variety?
e.g: $F(\{0\}$x $D)=\{(f(t),g(t)):t \in D\}$. where $f$ and $g$ are holomorphic on the closure of $D$.