$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$ $\newcommand{\sgn}{\operatorname{sgn}}$ $\newcommand{\rank}{\operatorname{rank}}$
Theorem 1. \begin{gather} P(z,w)= \sum_{j=0}^m a_j(w) z^j \end{gather} where $a_j(w)$ are analytic functions. Assume that $z^*$ is a root of multiplicity $r$ of $P(z, w^*)$. Then for $w$ in the vicinity of $w^*$ polynomial $P(z,w)$ has $q$ roots in the vicinity of $z^*$, and these roots are given by Puiseux series \begin{gather} z_k(w)= \sum_{n=0}^\infty b_{k,n} (w-w^*)^{\frac{n}{q_k}} \end{gather} where $q_1+\ldots +q_k=r$. Here $z_k$ is $q_k$-valued function in the vicinity of $w^*$.
Remark 1. Similar decomposition holds if $a_j(w)$ are given by Puiseux series in the vicinity of $w^*$.
Ciorollary 1. In the framework of Theorem 1 assume that for real $w$ in the vicity of $w^*$ all roots of $P(z,w)$ (which are in the vicinity of $z^*$) are real. Then $k=r$, $q_1=\ldots=q_k=1$ and $z_k(w)$ are analytic functions in the vicinity of $w^*$.
Let now $w\in \bR^d$ and $a_j(w)$ be real-valued functions. What we have instead Corollary 1?
There is rather difficult
Theorem 2. Let now $w\in \bR^d$ and $a_j(w)$ be real-valued functions. Assume that for real $w$ in the vicity of $w^*$ all roots of $P(z,w)$ (which are in the vicinity of $z^*$) are real: $z_1(w),\ldots, z_r(w)$.
Then directional derivatives are bounded: \begin{gather} |\ell\cdot \nabla z_k| \le C|\ell|. \end{gather}
Remark 2. $z_k$ are not necessarily $C^1$. Moreover, their higher-order derivatives are not necessarily bounded.
F.e. consider $P(z,w)= z^2-(w_1^2+ w_2^2)$. Then $z_{1,2}=\pm \sqrt{w_1^2+w_2^2}$.