Let $B_i$ be the interior of a polygon, and $B_o$ be the exterior of a polygon (both $B_i$ and $B_o$ are open subsets of the complex plane $\mathbb{C}$).
Let $p(z)$ be polynomial of degree $n$ with $0$ constant term. We are looking for a pair of functions $(\psi_o(z), \psi_i(z))$, where $\psi_o: \overline{B_o} \to \mathbb{C}$ and $\psi_i: \overline{B_i} \to \mathbb{C}$ are continuous functions which are holomorphic on the interiors of their domains, namely they are holomorphic on $B_o$ and $B_i$ respectively.
We are interesting in the following problem $(P)$:
Does there exist a pair of functions $(\psi_o, \psi_i)$ as above, such that
1) the principal part of $\psi_o$ at $\infty$ is equal to $p(z)$,
2) $\operatorname{Re}(\psi_o) = 0$ on the polygon $\partial B_i$, and
3) $\operatorname{Im}(\psi_o) = \operatorname{Im}(\psi_i)$ on the polygon $\partial B_i$.
This was the existence part, and my first question.
As to the uniqueness part, note that given a solution $(\psi_o,\psi_i)$ of $(P)$, one can add to that solution $(bi,a+bi)$ (in other words, one adds $bi$ to $\psi_o$ and adds $a+bi$ to $\psi_i$), where $a$ and $b$ are real numbers, and get another solution. Is the solution to $(P)$, assuming it exists, unique up to adding some $(bi,a+bi)$? This is my second question.