Let $m,n \in \mathbb{N}$, let $U \subseteq \mathbb{C}^m \times \mathbb{C}^n$ be open, let $(a,b) \in U$, and let $F: U \to \mathbb{C}^n$ be holomorphic. Write points of $\mathbb{C}^m \times \mathbb{C}^n$ as $(z,w)$ with $z \in \mathbb{C}^m$ and $w \in \mathbb{C}^n$. Suppose that $F(a,b)=0$ and that the complex $n \times n$ matrix

\begin{align*} \left(\frac{\partial F_i}{\partial w_j}(a,b)\right)_{1 \leq i,j \leq n} \end{align*}

is invertible. Then there exist open neighbourhoods $A \subseteq \mathbb{C}^m$ of $a$ and $B \subseteq \mathbb{C}^n$ of $b$, with $A \times B \subseteq U$, and a holomorphic map $g: A \to B$ such that

\begin{align*} \{(z,w) \in A \times B : F(z,w)=0\}=\{(z,g(z)) : z \in A\}. \end{align*}