Holomorphic factorization of polynomials

We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial $P(s,\lambda)$, $s\in\mathbb C^n$, $\lambda\in\mathbb C$, in a domain $\Omega\subset\mathbb C^n$ which is connected with the ordering of the real part of the roots of the equation $P(s,\lambda)=0$, $s\in\Omega$.