Let $\Omega$ be an open subset of $\mathbb R^d$, let $U$ be an open subset of $\mathbb C$ and let $f:\Omega\times U\rightarrow\mathbb C$ be a $C^\infty$ function which is holomorphic with respect to $\zeta$ on $U$. Let me assume that for some $(x_0,\zeta_0)\in \Omega\times U$, $$ f(x_0,\zeta_0)=0,\quad \frac{\partial f}{\partial \zeta}(x_0,\zeta_0)\not=0. $$ I want to show that there exists a smooth function $\lambda$ defined near $x_0$ such that $\lambda(x_0)=\zeta_0$ and $$ f(x,\zeta)=e_0(x,\zeta)(\zeta-\lambda(x)), $$ where $e_0$ is a $C^\infty$ non-vanishing function defined near $(x_0,\zeta_0)$ which is holomorphic with respect to $\zeta$ and $\lambda$ is a $C^\infty$ function defined near $x_0$.
This is obvious when $f$ is analytic by the holomorphic implicit function theorem. However, with $f$ only $C^\infty$ with respect to $x$, I would like to avoid resorting to sophisticated arguments such as the Malgrange Preparation Theorem.
