Lemma 9.2.5.C47992939550 Let $n\geq 1$, let $P(z',w)=w^\mu +a_1(z')w^{\mu -1}+\cdots +a_\mu (z')$ be a monic polynomial in $w$ whose coefficients are holomorphic near a point $x'\in \mathbb C^{n-1}$, and let $w_0$ be a root of multiplicity $\mu '\geq 1$ of $P(x',\cdot )$. Then, near $x=(x',w_0)$,
\[ P=P_ xQ_ x, \]
where $P_ x$ is a Weierstrass polynomial of degree $\mu '$ in $w-w_0$ with respect to the coordinates $(z'-x',w-w_0)$, and $Q_ x$ is a monic polynomial of degree $\mu -\mu '$ in $w$ with coefficients holomorphic near $x'$ and $Q_ x(x)\neq 0$.
Proof.
In the coordinates $(z'-x',w-w_0)$ centered at $x$, the function $w\mapsto P(x',w)$ has a zero of order $\mu '$ at $w_0$. By the Weierstrass preparation theorem (theorem 5.5.2), $P$ is, near $x$, a nowhere vanishing holomorphic function $U$ times a Weierstrass polynomial $P_ x$ of degree $\mu '$ in $w-w_0$. Since $P_ x$ is monic in $w$, Euclidean division of the polynomial $P$ by $P_ x$, with coefficients holomorphic near $x'$, gives $P=P_ xQ+R$, where $Q$ is a monic polynomial of degree $\mu -\mu '$ in $w$ and $R$ is a polynomial in $w$, equivalently in $w-w_0$, of degree less than $\mu '$. The identities $P=P_ xQ+R$ and $P=P_ xU+0$ are two divisions of the germ of $P$ at $x$ by $P_ x$; by the uniqueness in proposition 9.2.2, applied in the coordinates centered at $x$, we get $Q=U$ near $x$ and $R=0$. Hence $Q_ x=Q$ has the required properties, with $Q_ x(x)=U(x)\neq 0$.
$\square$
Pome
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