Let $f(w,z)=z^n+a_1(w)z^{n-1}+\cdots+a_n(w)$ be a Weierstrass-polynomial of degree $n$ in $\lambda$, whose coefficients are all analytic functions of $w\in\mathbb C^l$. Suppose that $z=z_0$ is a root with multiplicity $m$ of $f(w_0,z)=0$.
Q Can we say that for $w$ near $w_0$, there are exactly $m$ roots $z(w)$ of $f(w,z)$ near $z_0$, and these roots $z(w)$ are the branches of one or more multivalued analytic functions with at worst algebraic branch points at $z=z_0$?