Let $n\ge 1$, let $M\subset \mathbb C^{n+1}$ be a real-analytic embedded real hypersurface, and let $p\in M$. Assume that $M$ is strictly pseudoconvex at $p$, meaning that, after choosing the pseudoconvex side, the Levi form of a $C^2$ local defining function is positive definite on $T_pM\cap iT_pM$. Then there exist an open neighbourhood $U\subset\mathbb C^{n+1}$ of $p$ and a biholomorphic map $\Phi:U\to \Phi(U)\subset\mathbb C^n\times\mathbb C$ such that $\Phi(p)=0$ and, writing the target coordinates as $(z,w)=(z_1,\dots,z_n,u+iv)$, the hypersurface germ $\Phi(M\cap U)$ at $0$ is given by a convergent real-analytic equation

\begin{align*} v=|z|^2+\sum_{k,l\ge 2}F_{kl}(z,\bar z,u). \end{align*}

Here $|z|^2=\sum_{j=1}^n |z_j|^2$, and each coefficient $F_{kl}:\mathbb C^n\times\mathbb C^n\times\mathbb R\to\mathbb R$ is real-analytic near $(0,0,0)$, homogeneous of degree $k$ in $z$, and homogeneous of degree $l$ in $\bar z$.

For a polynomial $P(z,\bar z,u)$ in $z$ and $\bar z$ with real-analytic dependence on $u$, define the Chern-Moser trace operator by

\begin{align*} \operatorname{tr}P=\sum_{j=1}^n \frac{\partial^2 P}{\partial z_j\,\partial \bar z_j}. \end{align*}

Let $\operatorname{tr}^m$ denote the $m$-fold iterate of $\operatorname{tr}$. The coefficients satisfy the Chern-Moser trace conditions

\begin{align*} \operatorname{tr}F_{22}=0, \end{align*}

\begin{align*} \operatorname{tr}^2F_{23}=0, \end{align*}

\begin{align*} \operatorname{tr}^3F_{33}=0. \end{align*}

Let

\begin{align*} Q=\{(z,w)\in\mathbb C^n\times\mathbb C:\operatorname{Im}(w)=|z|^2\}. \end{align*}

If $\Phi_1$ and $\Phi_2$ are two biholomorphic coordinate germs at $p$ satisfying the preceding normal-form conclusions, then the transition germ $\Phi_2\circ\Phi_1^{-1}$ is the germ at $0$ of a biholomorphic automorphism of $Q$ fixing $0$. Conversely, composing any normalizing coordinate germ on the left with an element of $\operatorname{Aut}(Q)_0$ gives another admissible normalizing coordinate germ exactly when the resulting equation is again in the displayed Chern-Moser normal form. Thus the residual freedom in normalized coordinates is precisely the isotropy action of $\operatorname{Aut}(Q)_0$; after fixing one representative of this isotropy, the normalizing coordinate germ and the coefficients $F_{kl}$ are unique.