If $p,q$ are polynomials and $q$ divides $p$ in the space of holomorphic functions, then the quotient is a polynomial

Question

Let $\Omega$ be a bounded domain in $\mathbb{C}^{m}$. $p,q$ are two polynomials in $\mathbb{C}[z_{1},\ldots,z_{m}]$ and $q$ divides $p$ in $\mathcal{O}(\Omega)$. Then $p=fq$, for some $f\in\mathcal{O}(\Omega)$. If we assume that $Z(q)\cap\Omega$ is non-empty, then is it true that $f$ is a polynomial?

Answer 1

No.

For example, take $m=1$, $\Omega=\mathbb{D}$, $q=z(z+2)$ and $p=z$. Then $Z(q)\cap\Omega=\{0\}$, $f=\frac1{z+2}$ is holomorphic on $\Omega$ but not a polynomial.